Content-Type: multipart/mixed; boundary="-------------0012311444579" This is a multi-part message in MIME format. ---------------0012311444579 Content-Type: text/plain; name="00-521.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-521.keywords" C*-algebra, crossed product, spectral theory, essential spectrum, Mourre estimate, N-body problem, anisotropic systems, quantum fields, Klaus type potentials ---------------0012311444579 Content-Type: application/postscript; name="caeoI.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="caeoI.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: caeoI.dvi %%Pages: 73 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips caeoI.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.12.31:2110 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict 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b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.) g(.)g(.)138 b(8)615 4403 y(1.7)124 b(Notations)80 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f (.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(10)p 448 4465 1196 4 v 551 4519 a Fx(\003)587 4550 y Fw(CNRS)d(and)1103 4519 y Fv(?)1139 4550 y Fw(Departmen)n(t)f(of)j(Mathematics,)107 b(Univ)n(ersit)n(y)89 b(of)i(Cergy-P)n(on)n(toise)448 4642 y(2,)96 b(a)n(v)n(en)n(ue)81 b(Adolphe)g(Chauvin)g(95302)i (Cergy-P)n(on)n(toise)g(Cedex,)96 b(F)-6 b(rance,)96 b Fu(e-mail:)448 4733 y(Vladimir.Georgescu@math.u-ce)q(rgy.)q(fr)45 b Fw(and)39 b Fu(Andrei.Iftimovici@math.u-cer)q(gy.f)q(r)1920 5225 y Fy(1)p eop %%Page: 2 2 2 1 bop 448 573 a Fz(2)85 b(Observ)-6 b(ables)35 b(and)g(their)f(Essen) m(tial)h(Sp)s(ectrum)831 b(11)615 686 y Fy(2.1)124 b(Observ)-5 b(ables)29 b(a\016liated)h(to)h Ft(C)1915 653 y FG(\003)1954 686 y Fy(-algebras)72 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)g(.)92 b(11)615 799 y(2.2)124 b(Self-adjoin)m(t)30 b(op)s(erators)g(a\016liated)g(to)h Ft(C)2296 766 y FG(\003)2335 799 y Fy(-algebras)115 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)92 b(11)615 912 y(2.3)124 b(A\016liation:)40 b(a)30 b(general)h(criterion) 56 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g (.)92 b(12)615 1024 y(2.4)124 b(Computation)30 b(of)g Fs(J)p Fy(-)q Fr(\033)1653 1038 y Fq(ess)1744 1024 y Fy(\()p Ft(H)7 b Fy(\))130 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(13)615 1137 y(2.5)124 b(Restricted)31 b(pro)s(ducts)e(of)h Ft(C)1843 1104 y FG(\003)1882 1137 y Fy(-algebras)73 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.) h(.)g(.)f(.)h(.)g(.)g(.)92 b(15)615 1250 y(2.6)124 b(Asymptotic)31 b(algebras)75 b(.)45 b(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(15)448 1454 y Fz(3)85 b(Crossed)35 b(Pro)s(ducts)1941 b(17)615 1567 y Fy(3.1)124 b(De\014nition)29 b(of)i(crossed)f(pro)s(ducts)54 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.) 92 b(17)615 1680 y(3.2)124 b(F)-8 b(unctorial)30 b(prop)s(erties)55 b(.)45 b(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)g(.)92 b(18)615 1793 y(3.3)124 b(Crossed)30 b(pro)s(ducts)f(of)i(direct)e(pro)s(ducts)100 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(20)615 1906 y(3.4)124 b(P)m(osition)30 b(and)g(momen)m(tum)g(observ)-5 b(ables)70 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(21)615 2019 y(3.5)124 b(Crossed)30 b(pro)s(ducts)f(and)h(pseudo)s (di\013eren)m(tial)d(op)s(erators)e(.)46 b(.)f(.)h(.)g(.)g(.)92 b(23)615 2132 y(3.6)124 b(Three)30 b(descriptions)f(of)h Fp(K)15 b Fy(\()p Ft(X)8 b Fy(\))87 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)g(.) f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(25)615 2245 y(3.7)124 b(Auxiliary)28 b(results)86 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f (.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(26)448 2448 y Fz(4)85 b(Algebras)35 b(of)g(hamiltonians:)46 b(examples)1115 b(27)615 2561 y Fy(4.1)124 b(General)31 b(considerations)57 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.) h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(27)615 2674 y(4.2)124 b(Lo)s(calizations)30 b(at)h(in\014nit)m(y)85 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.) h(.)g(.)g(.)92 b(29)615 2787 y(4.3)124 b(T)-8 b(ranslation)29 b(in)m(v)-5 b(arian)m(t)30 b(\014lters)84 b(.)45 b(.)h(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(32)615 2900 y(4.4)124 b(The)30 b(Loren)m(tz)i(\014lter)111 b(.)46 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)g(.)92 b(33)615 3013 y(4.5)124 b(One)30 b(dimensional)e(anisotrop)m(y)65 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)g(.)f (.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(35)615 3126 y(4.6)124 b(Graded)30 b(algebras)100 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f (.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(38)615 3239 y(4.7)124 b(Quan)m(tum)30 b(\014elds)67 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(40)448 3443 y Fz(5)85 b(Bumps)34 b(Algebras)1994 b(41)615 3555 y Fy(5.1)124 b(The)30 b(algebra)h(of)f(classical)g(in)m(teractions)j(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(41)615 3668 y(5.2)124 b(Bumps)30 b(algebras)g(and)g(a)h(HVZ)f (theorem)66 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(46)615 3781 y(5.3)124 b(Dense)31 b(subalgebras)90 b(.)46 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)g(.)92 b(50)615 3894 y(5.4)124 b(Phase-space)32 b(description)c(of)j(bumps)d(algebras)44 b(.)i(.)g(.)f(.)h(.)g(.)f(.)h (.)g(.)g(.)92 b(51)615 4007 y(5.5)124 b(A\016liation)29 b(and)h(a)h(generalized)f(Klaus)f(theorem)78 b(.)46 b(.)f(.)h(.)g(.)f (.)h(.)g(.)g(.)92 b(57)615 4120 y(5.6)124 b(Scattering)62 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.) g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(61)448 4324 y Fz(6)85 b(App)s(endix)2302 b(63)615 4437 y Fy(6.1)124 b(En)m(v)m(eloping)30 b Ft(C)1395 4404 y FG(\003)1434 4437 y Fy(-Algebras)74 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(63)615 4550 y(6.2)124 b(Pro)s(of)30 b(of)h(Prop)s(osition)d(3.8)74 b(.)46 b(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(65)615 4663 y(6.3)124 b(W)-8 b(a)m(v)m(e)33 b(op)s(erators)61 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)92 b(65)483 4866 y Fz(References)2360 b(69)1920 5225 y Fy(2)p eop %%Page: 3 3 3 2 bop 448 573 a FA(1)135 b(In)l(tro)t(duction)448 854 y Fz(1.1.)159 b Fy(The)43 b Ft(C)1041 821 y FG(\003)1080 854 y Fy(-algebra)g(generated)i(b)m(y)e(the)g(\\admissible")e (hamiltonians)g(of)j(a)448 967 y(quan)m(tum)d(system)f(often)i(has)e(a) h(remark)-5 b(able)40 b(algebraic)h(structure)f(whic)m(h)g(allo)m(ws) 448 1080 y(one)33 b(to)g(compute)g(explicitly)c(its)j(quotien)m(t)h (with)d(resp)s(ect)j(to)g(the)f(ideal)f(of)i(compact)448 1193 y(op)s(erators)k(\(what)g(w)m(e)g(mean)f(b)m(y)g(admissible)e (will)g(b)s(ecome)j(clear)f(later)h(on\).)59 b(This)448 1306 y(statemen)m(t)34 b(should)29 b(b)s(e)i(considered)f(as)i(an)f (\\exp)s(erimen)m(tal")g(observ)-5 b(ation:)43 b(our)31 b(aim)448 1419 y(is)c(to)i(justify)d(it)h(b)m(y)h(examples)f(and)h(to)g (sho)m(w)g(that)g(it)g(has)f(in)m(teresting)g(consequences)448 1532 y(in)36 b(the)h(sp)s(ectral)f(theory)h(of)f(the)h(hamiltonians)e (of)h(ph)m(ysical)g(systems)g(with)g(a)h(ric)m(h)448 1644 y(in)m(ternal)44 b(structure)g(\(lik)m(e)h(man)m(y)g(b)s(o)s(dy)e (or)i(quan)m(tum)f(\014eld)g(systems\))h(or)g(a)g(high)448 1757 y(degree)30 b(of)f(anisotrop)m(y)g(in)f(con\014guration)h(or)g (phase)g(space.)41 b(W)-8 b(e)30 b(ha)m(v)m(e)g(in)e(mind)f(here)448 1870 y(only)h(sp)s(ectral)f(prop)s(erties)g(whic)m(h)g(are)i(stable)f (under)f(\(certain)h(classes)h(of)7 b(\))29 b(compact)448 1983 y(p)s(erturbations:)44 b(for)33 b(the)g(momen)m(t)h(w)m(e)f(are)h (concerned)f(only)f(with)g(the)h(description)448 2096 y(of)e(the)f(essen)m(tial)h(sp)s(ectrum)e(and)h(the)g(pro)s(of)g(of)g (the)h(Mourre)f(estimate.)589 2209 y(Instead)46 b(of)f(fo)s(cusing)f (on)h(the)h(study)e(of)h(one)h(self-adjoin)m(t)f(op)s(erator)g Ft(H)52 b Fy(\()p Fo(the)448 2322 y Fy(hamiltonian\),)26 b(w)m(e)i(consider)e(as)i(primary)d(ob)5 b(ject)28 b(of)g(the)f(theory) g(the)h(ric)m(her)e(mathe-)448 2435 y(matical)e(structure)g(\()p Fn(H)j Ft(;)15 b Fs(C)p Fy(\))25 b(consisting)e(of)h(a)h(Hilb)s(ert)d (space)i Fn(H)51 b Fy(\(the)25 b(state)g(space)g(of)448 2548 y(the)g(quan)m(tum)e(system\))i(and)e(a)i Ft(C)1619 2515 y FG(\003)1657 2548 y Fy(-subalgebra)f Fs(C)g Fy(of)g Ft(B)5 b Fy(\()p Fn(H)27 b Fy(\),)e(the)g Fo(algebr)-5 b(a)28 b(of)f(ener)-5 b(gy)448 2661 y(observables)7 b Fy(,)33 b(or)f Fo(algebr)-5 b(a)34 b(of)g(hamiltonians)p Fy(.)46 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Fy(of)36 b Ft(C)3368 3921 y FG(\003)3407 3954 y Fy(-)448 4067 y(algebras)31 b Fs(C)864 4081 y FF(i)922 4067 y Fy(equipp)s(ed)d(with)h(ideals)g Fs(J)1826 4081 y FF(i)1885 4067 y Fy(and)g(there)i(is)e(a)i(natural)f (em)m(b)s(edding)448 4286 y(\(1.3\))1294 4251 y Fs(C)1340 4185 y Fm(.)1399 4329 y Ft(K)7 b Fy(\()p Fn(H)26 b Fy(\))1724 4286 y Ft(,)-15 b Fl(!)1881 4183 y Fm(Q)1967 4278 y FF(i)p FG(2)p FF(I)2093 4251 y Fs(C)2154 4265 y FF(i)2167 4185 y Fm(.)2225 4261 y(L)2326 4356 y FF(i)p FG(2)p FF(I)2453 4329 y Fs(J)2508 4343 y FF(i)2566 4286 y Ft(:)448 4525 y Fy(Note)37 b(that)f(in)f(the)h(righ)m(t)f(hand)f(side)h(w)m(e)h(ha)m (v)m(e)h(b)s(oth)e(a)h(direct)f(pro)s(duct)3058 4457 y Fm(Q)3144 4552 y FF(i)p FG(2)p FF(I)3291 4525 y Fy(and)448 4638 y(a)41 b(direct)e(sum)1003 4569 y Fm(L)1104 4664 y FF(i)p FG(2)p FF(I)1255 4638 y Fy(of)h Ft(C)1440 4605 y FG(\003)1479 4638 y Fy(-algebras.)70 b(The)40 b(preceding)f(case)i (is)e(obtained)g(when)448 4751 y Fs(J)503 4765 y FF(i)578 4751 y Fy(=)46 b 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Ft(H)49 b Fy(is,)28 b(in)f(the)h(represen)m (tation)g(giv)m(en)g(b)m(y)g(the)448 916 y(righ)m(t)37 b(hand)e(side)h(of)h(\(1.3\),)j(the)d(quotien)m(t)g(of)g(some)g(family) e Fl(f)p Ft(H)2734 930 y FF(i)2762 916 y Fl(g)2807 930 y FF(i)p FG(2)p FF(I)2955 916 y Fy(of)i(op)s(erators)448 1029 y Ft(H)524 1043 y FF(i)582 1029 y Fy(a\016liated)30 b(to)h Fs(C)1128 1043 y FF(i)1156 1029 y Fy(,)g(then)448 1212 y(\(1.4\))256 b Fr(\033)943 1226 y Fq(ess)1034 1212 y Fy(\()p Ft(H)7 b Fy(\))26 b(=)1350 1126 y Fm(\\)1338 1320 y Fv(F)8 b Fx(\032)p Fv(I)1309 1368 y(F)g Fk(\014nite)1508 1111 y Fm(n)1569 1139 y(\002)1607 1144 y(S)1682 1239 y FF(i)p FG(2)p FF(F)1827 1212 y Fr(\033)1881 1226 y Fq(ess)1972 1212 y Fy(\()p Ft(H)2083 1226 y FF(i)2111 1212 y Fy(\))2146 1139 y Fm(\003)2205 1212 y Fl([)2286 1111 y Fm(h)p 2329 1133 525 4 v 33 x(S)2404 1239 y FF(i)p FG(2)p FF(I)d FG(n)p FF(F)2620 1212 y Fr(\033)l Fy(\()p Ft(H)2789 1226 y FF(i)2817 1212 y Fy(\))2853 1111 y Fm(io)2971 1212 y Ft(:)448 1501 y 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b(considered)f(as)i(a)f(\014xed)g(ob)5 b(ject)448 2483 y(and,)24 b(in)d(the)h(simplest)e(cases,)25 b(its)d(sp)s(ectral)f(analysis)g(is)g(p)s(erformed)g(in)g(a)h(p)s (erturbativ)m(e)448 2596 y(manner:)38 b(a)28 b(second)f(op)s(erator) 1540 2573 y Fm(c)1536 2596 y Ft(H)49 b Fy(\(the)28 b(\\free)f (hamiltonian",)f(usually)f(denoted)i Ft(H)3363 2610 y Fq(0)3402 2596 y Fy(\))448 2718 y(with)h(kno)m(wn)g(sp)s(ectral)g(prop) s(erties)f(is)h(giv)m(en)h(suc)m(h)f(that)h(\()p Ft(H)24 b Fl(\000)17 b Ft(z)t Fy(\))2734 2685 y FG(\000)p Fq(1)2846 2718 y Fl(\000)f Fy(\()2972 2695 y Fm(c)2968 2718 y Ft(H)40 b Fl(\000)16 b Ft(z)t Fy(\))3252 2685 y FG(\000)p Fq(1)3376 2718 y Fy(is)448 2831 y(compact.)53 b(F)-8 b(rom)35 b(this)d(one)j (deduces)e(that)i Ft(H)41 b Fy(has)33 b(sp)s(ectral)h(prop)s(erties)e (similar)f(to)448 2944 y(those)23 b(of)777 2921 y Fm(c)774 2944 y Ft(H)f Fy(.)38 b(In)21 b(other)i(terms,)h(one)f(obtains)e(a)i (simpler)d(op)s(erator)2752 2921 y Fm(c)2748 2944 y Ft(H)45 b Fy(b)m(y)22 b(subtracting)448 3057 y(\(in)30 b(a)g(generalized)g (sense\))h(from)f Ft(H)37 b Fy(a)31 b(compact)h(op)s(erator.)589 3170 y(Unfortunately)-8 b(,)43 b(in)c(more)h(complicated)g(situations)f (\(e.g.)j(in)d(the)h(three)h(b)s(o)s(dy)448 3283 y(problem\))27 b Ft(H)34 b Fy(do)s(es)27 b(not)h(b)s(ecome)g(signi\014can)m(tly)d (simpler)h(after)i(the)f(subtraction)g(of)h(a)448 3396 y(compact)i(op)s(erator.)41 b(So)28 b(among)i(the)f(op)s(erators)g (acting)g(in)e Fn(H)56 b Fy(there)29 b(is)e(no)i(natural)448 3509 y(candidate)40 b(for)1025 3486 y Fm(c)1022 3509 y Ft(H)22 b Fy(.)70 b(In)40 b(order)g(to)h(b)m(ypass)f(this)f(problem)g (in)g(concrete)j(mo)s(dels)d(of)448 3621 y(the)f Ft(N)10 b Fy(-b)s(o)s(dy)35 b(problem,)j(quan)m(tum)e(\014eld)g(theory)-8 b(,)39 b(or)e(solid)f(state)i(ph)m(ysics,)g(sev)m(eral)448 3734 y(metho)s(ds)27 b(ha)m(v)m(e)i(b)s(een)e(in)m(v)m(en)m(ted.)40 b(F)-8 b(or)29 b(example,)f(the)g(partitions)e(of)i(unit)m(y)f(in)f (con\014g-)448 3847 y(uration)35 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b(think,)e(ho)m(w)m(ev)m(er,)i(that)f(the)g (approac)m(h)g(dev)m(elop)s(ed)f(in)f(this)h(pap)s(er)g(is)f(less)h(ad) 448 2398 y(ho)s(c)41 b(and)e(has)i(certain)f(conceptual)g(and)g(tec)m (hnical)h(adv)-5 b(an)m(tages)42 b(with)d(resp)s(ect)h(to)448 2511 y(other)27 b(metho)s(ds.)39 b(Its)27 b(main)f(feature)h(is)f(that) h(it)g(allo)m(ws)f(one)h(to)h(go)f(b)s(ey)m(ond)f(the)i(usual)448 2624 y(p)s(erturbativ)m(e)h(tec)m(hniques)g(\(ev)m(en)i(in)e(their)g (re\014ned)g(forms)g(in)m(v)m(olving)f(v)-5 b(arious)29 b(kinds)448 2746 y(of)h(partitions)e(of)i(unit)m(y\))f(in)f(a)i(simple) d(and)i(natural)g(w)m(a)m(y)-8 b(.)41 b(The)29 b(co-hamiltonian)3317 2723 y Fm(c)3314 2746 y Ft(H)22 b Fy(,)448 2859 y(canonically)k(asso)s (ciated)h(to)h(the)f(hamiltonian)e(of)i(the)g(system,)h(pla)m(ys)e(an)h (imp)s(ortan)m(t)448 2972 y(role)j(not)h(only)e(in)h(the)g(sp)s(ectral) g(theory)g(of)h Ft(H)7 b Fy(,)30 b(but)g(also)g(in)f(its)h(scattering)h (theory)-8 b(.)589 3085 y(There)25 b(is)f(a)i(v)m(ery)g(large)f (literature)f(on)h(the)h(applications)d(of)i(the)h(abstract)g(theory) 448 3198 y(of)e Ft(C)617 3165 y FG(\003)656 3198 y Fy(-algebras)f(to)h (the)g(study)e(of)i(the)f(sp)s(ectral)g(prop)s(erties)e(of)j(v)-5 b(arious)22 b(classes)h(of)h(op-)448 3311 y(erators.)40 b(One)26 b(ma)m(y)g(\014nd)f(references)h(on)g(this)f(question)g(in)g ([Da)m(v)q(,)h(Dou)q(,)g(Mur)o(,)h(W)-8 b(e-O)q(])448 3423 y(for)41 b(example)g(\(in)f(particular,)i(the)g(theory)f(of)g (extensions)g(of)g Ft(K)7 b Fy(\()p Fn(H)27 b Fy(\))41 b(is)f(dev)m(oted)448 3536 y(to)30 b(the)f(classi\014cation)f(of)i(the) f(algebras)g(ha)m(ving)f(certain)h(quotien)m(ts)h(with)d(resp)s(ect)i (to)448 3649 y Ft(K)7 b Fy(\()p Fn(H)27 b Fy(\)\).)39 b(But)25 b(our)f(purp)s(oses)f(and)h(tec)m(hniques)g(here)g(are)h (quite)f(di\013eren)m(t.)39 b(Closer)23 b(to)448 3762 y(ours)28 b(is)f(the)i(p)s(oin)m(t)e(of)h(view)g(of)g(H.)h(O.)f(Cordes) f([Cor];)j(but)d(his)g(goal,)j(the)e(class)g(of)g(ex-)448 3875 y(amples)g(he)g(considers,)g(and)f(his)g(metho)s(ds)h(are)g(v)m (ery)h(far)f(from)g(ours.)39 b(The)28 b(approac)m(h)448 3988 y(to)37 b(the)g Ft(N)10 b Fy(-b)s(o)s(dy)35 b(problem)f(exp)s (osed)i(in)f(the)h(c)m(hapters)h(8)g(and)e(9)i(from)f([ABG)q(])g(\(see) 448 4101 y(also)23 b(references)h(therein\))e(should)f(b)s(e)h (considered)g(as)h(an)g(example)g(of)g(the)g(tec)m(hniques)448 4214 y(dev)m(elop)s(ed)30 b(in)f(the)i(presen)m(t)f(pap)s(er.)448 4405 y Fz(1.5.)87 b Fy(The)31 b(\014rst)f(problem)g(one)h(has)g(to)h (solv)m(e)g(in)e(the)h(algebraic)g(approac)m(h)h(that)g(w)m(e)448 4518 y(use)f(is)f(the)h(c)m(hoice)h(of)f(the)g Ft(C)1459 4485 y FG(\003)1498 4518 y Fy(-algebra)h(of)f(energy)g(observ)-5 b(ables)30 b(of)h(a)h(giv)m(en)f(system:)448 4631 y(one)i(has)f(to)i (isolate)e(the)h(class)f(of)g(self-adjoin)m(t)g(op)s(erators)h(whic)m (h)e(are)i(natural)e(can-)448 4744 y(didates)k(as)i(hamiltonians)c(of)j (the)g(system)g(in)e(a)j(giv)m(en)f(ph)m(ysical)e(situation.)56 b(Then)448 4857 y Fs(C)33 b Fy(will)c(b)s(e)j(the)h Ft(C)1072 4824 y FG(\003)1111 4857 y Fy(-algebra)f(generated)i(b)m(y)e(the)h (resolv)m(en)m(ts)g(of)f(these)h(op)s(erators.)47 b(A)448 4970 y(priori)22 b(this)h(do)s(es)h(not)h(seem)f(to)h(giv)m(e)g(a)g (nice)e(mathematical)i(ob)5 b(ject.)39 b(The)24 b(core)h(of)g(our)1920 5225 y(7)p eop %%Page: 8 8 8 7 bop 448 573 a Fy(pap)s(er)26 b(is)g(to)h(sho)m(w)g(that,)h(on)f (the)f(con)m(trary)-8 b(,)29 b(one)e(often)g(gets)h(in)d(this)h(w)m(a)m (y)i(in)m(teresting)448 686 y(classes)35 b(of)f Ft(C)921 653 y FG(\003)960 686 y Fy(-algebras.)53 b(This)33 b(question)h(will)d (b)s(e)j(discussed)e(in)i(detail)f(in)g(Section)448 799 y(4;)h(here)e(w)m(e)h(shall)d(only)h(p)s(oin)m(t)g(out)i(the)f(role)g (of)g Ft(C)2219 766 y FG(\003)2258 799 y Fy(-crossed)h(pro)s(ducts)e (in)f(this)h(con-)448 912 y(text)g(\(crossed)f(pro)s(ducts)e(and)h (other)h(notions)f(whic)m(h)g(are)h(not)g(standard)f(for)g(p)s(eople) 448 1024 y(w)m(orking)h(in)f(sp)s(ectral)h(theory)g(will)e(b)s(e)i(in)m (tro)s(duced)f(in)g(the)h(b)s(o)s(dy)f(of)i(the)f(pap)s(er\).)589 1137 y(Let)j Fn(A)54 b Fy(b)s(e)31 b(a)i(closed)f Fl(\003)p Fy(-subalgebra)g(of)g(the)h(algebra)f Ft(C)2543 1104 y Fq(u)2536 1165 y(b)2586 1137 y Fy(\()p Fp(R)2681 1104 y FF(n)2734 1137 y Fy(\))h(of)f(b)s(ounded)f(uni-)448 1250 y(formly)f(con)m(tin)m(uous)g(functions)g(on)g Fp(R)1771 1217 y FF(n)1824 1250 y Fy(.)42 b(Assume)31 b(that)g Fn(A)52 b Fy(con)m(tains)31 b(the)g(constan)m(ts)448 1363 y(and)21 b(is)g(stable)g(under)f(translations)g(and)h(let)g(us)g (denote)h Fn(A)2465 1330 y FG(1)2561 1363 y Fy(the)g(set)g(of)g Ft(')j Fl(2)g Ft(C)3179 1330 y FG(1)3253 1363 y Fy(\()p Fp(R)3349 1330 y FF(n)3402 1363 y Fy(\))448 1476 y(suc)m(h)g(that)h Ft(')899 1443 y Fq(\()p FF(\013)p Fq(\))1029 1476 y Fl(2)e Fn(A)47 b Fy(for)24 b(all)g Ft(\013)p Fy(.)40 b(Let)25 b(\001)g(=)g Ft(P)2036 1443 y Fq(2)2100 1476 y Fy(b)s(e)g(the)g(\(p)s (ositiv)m(e\))g(Laplace)g(op)s(erator)448 1589 y(and)38 b(let)f(us)h(consider)e(self-adjoin)m(t)i(op)s(erators)g(of)g(the)g (form)f Ft(H)45 b Fy(=)37 b(\001)25 b(+)g Ft(V)c Fy(,)40 b(where)448 1702 y Ft(V)58 b Fy(is)37 b(a)g(\014rst)g(order)g (di\013eren)m(tial)f(op)s(erator)i(with)e(co)s(e\016cien)m(ts)i(in)e Fn(A)2905 1669 y FG(1)2979 1702 y Fy(.)62 b(Then)37 b(the)448 1815 y(smallest)27 b Ft(C)867 1782 y FG(\003)905 1815 y Fy(-algebra)h(of)f(op)s(erators)h(on)f Ft(L)1936 1782 y Fq(2)1975 1815 y Fy(\()p Fp(R)2070 1782 y FF(n)2123 1815 y Fy(\))h(whic)m(h)e(con)m(tains)h(the)g(resolv)m(en)m(ts)h(of)448 1928 y(the)c(op)s(erators)g Ft(H)30 b Fy(is)23 b(equal)g(to)i(the)e (norm)g(closed)h(subspace)f(generated)h(b)m(y)g(op)s(erators)448 2041 y(of)30 b(the)g(form)f Ft(')p Fy(\()p Ft(Q)p Fy(\))p Ft( )s Fy(\()p Ft(P)13 b Fy(\),)33 b(where)c Ft(')d Fl(2)e Fn(A)51 b Fy(and)29 b Ft( )g Fl(2)c Ft(C)2354 2055 y Fq(0)2393 2041 y Fy(\()p Fp(R)2489 2008 y FF(n)2542 2041 y Fy(\))30 b(\(space)g(of)g(con)m(tin)m(uous,)448 2154 y(con)m(v)m(ergen)m(t)d(to)e(zero)g(at)g(in\014nit)m(y)c(functions\).) 38 b(No)m(w)25 b(this)e(algebra)h(is)f(just)h(the)g(crossed)448 2267 y(pro)s(duct)30 b Fn(A)42 b Fj(o)20 b Fp(R)1056 2234 y FF(n)1140 2267 y Fy(of)31 b Fn(A)52 b Fy(b)m(y)30 b(the)h(natural)f(action)h(of)g(the)g(translation)f(group)h(on)f(it.) 448 2379 y(This)36 b(is,)k(in)c(fact,)41 b(a)d(particular)f(case)i (\(see)f([DaG1)r(]\))h(of)f(Prop)s(osition)e(3.16,)41 b(whic)m(h)448 2492 y(justi\014es)31 b(the)h(in)m(terpretation)g(of)g (a)g(general)h(class)e(of)i(crossed)f(pro)s(ducts)f Fn(A)42 b Fj(o)21 b Ft(X)40 b Fy(as)448 2605 y(algebras)31 b(of)f(energy)h (observ)-5 b(ables.)589 2718 y(Ph)m(ysically)48 b(sp)s(eaking,)53 b Fn(A)70 b Fy(is)48 b(the)h(algebra)g(of)h(p)s(oten)m(tial)e(energies) h(that)h(w)m(e)448 2831 y(consider)41 b(as)h(natural)f(for)g(the)h(giv) m(en)g(system.)75 b(In)40 b(the)i(t)m(w)m(o)h(b)s(o)s(dy)e(case)h(w)m (e)g(tak)m(e)448 2944 y Fn(A)47 b Fy(=)25 b Ft(C)729 2958 y FG(1)803 2944 y Fy(\()p Fp(R)899 2911 y FF(n)952 2944 y Fy(\))k(\(con)m(tin)m(uous)h(functions)e(whic)m(h)g(ha)m(v)m(e)i (a)g(limit)d(at)j(in\014nit)m(y\).)39 b(F)-8 b(or)30 b(the)448 3057 y(simplest)22 b(t)m(yp)s(e)i(of)f(anisotrop)m(y)h(in)e (one)i(dimension)d(w)m(e)j(tak)m(e)h Fn(A)46 b Fy(=)25 b Ft(C)t Fy(\()p 2847 2983 66 4 v Fp(R)10 b Fy(\),)25 b(the)f(algebra)448 3170 y(of)35 b(con)m(tin)m(uous)f(functions)g(whic) m(h)f(ha)m(v)m(e)j(limits)c(\(distinct)i(in)f(general\))i(at)g(+)p Fl(1)g Fy(and)448 3283 y Fl(\0001)p Fy(.)54 b Fn(A)h Fy(could)34 b(b)s(e)g(an)h(algebra)g(of)g(p)s(erio)s(dic)d(or)i(almost) h(p)s(erio)s(dic)d(functions,)j(etc.)448 3396 y(A)g(treatmen)m(t)h(of)f (the)f Ft(N)10 b Fy(-b)s(o)s(dy)34 b(problem)e(from)i(this)g(p)s(oin)m (t)f(of)i(view)f(can)h(b)s(e)e(found)448 3509 y(in)c([DaG1)r(].)589 3621 y(It)g(is)e(imp)s(ortan)m(t)g(to)i(notice)g(that)f(ab)s(o)m(v)m(e) i(w)m(e)e(started)h(with)e(a)i(rather)f(small)e(alge-)448 3734 y(bra)32 b(of)h(con)m(tin)m(uous)f(functions)e Fn(A)54 b Fy(and)32 b(w)m(e)g(considered)f(only)h(\001)g(as)g(kinetic)g(energy) 448 3847 y(op)s(erator.)46 b(But)32 b(the)g(class)f(of)h(op)s(erators)g Ft(H)39 b Fy(a\016liated)31 b(to)i Fn(A)42 b Fj(o)21 b Fp(R)2788 3814 y FF(n)2873 3847 y Fy(is)30 b(m)m(uc)m(h)i(larger)448 3960 y(than)37 b(one)g(w)m(ould)e(exp)s(ect)j(at)f(\014rst)f(sigh)m(t.) 60 b(This)35 b(follo)m(ws)h(from)g(the)h(a\016liation)f(cri-)448 4073 y(terion)g(pro)m(v)m(ed)g(in)f([DaG2)q(].)58 b(In)35 b(the)h(t)m(w)m(o)h(b)s(o)s(dy)e(case)i(for)e(example,)i(a)g (self-adjoin)m(t)448 4186 y(op)s(erator)27 b Ft(H)33 b Fy(on)26 b Ft(L)1104 4153 y Fq(2)1143 4186 y Fy(\()p Fp(R)1238 4153 y FF(n)1291 4186 y Fy(\))h(is)e(strictly)g(a\016liated)g (to)i Fn(A)33 b Fj(o)12 b Fp(R)2471 4153 y FF(n)2550 4186 y Fy(if)25 b(and)h(only)f(if)g(there)i(is)e(a)448 4299 y(real)e(con)m(tin)m(uous)g(div)m(ergen)m(t)g(function)f Ft(h)h Fy(on)g Fp(R)2058 4266 y FF(n)2134 4299 y Fy(suc)m(h)g(that)h (\()p Ft(H)13 b Fy(+)6 b Ft(i)p Fy(\))2789 4266 y FG(\000)p Fq(1)2888 4299 y Fl(\000)g Fy(\()p Ft(h)p Fy(\()p Ft(P)13 b Fy(\))6 b(+)g Ft(i)p Fy(\))3342 4266 y FG(\000)p Fq(1)448 4412 y Fy(is)30 b(a)g(compact)i(op)s(erator.)448 4603 y Fz(1.6.)80 b Fy(W)-8 b(e)29 b(giv)m(e)g(no)m(w)f(a)g(short)g (description)e(of)i(the)g(con)m(ten)m(t)i(of)e(the)g(pap)s(er.)39 b(W)-8 b(e)29 b(ha)m(v)m(e)448 4716 y(c)m(hosen)d(to)f(study)f (\(except)i(for)f(examples\))f(systems)h(ha)m(ving)f(as)h (con\014guration)f(space)448 4829 y(an)j(arbitrary)f(ab)s(elian)f(lo)s (cally)h(compact)i(group)e Ft(X)7 b Fy(.)40 b(This)25 b(is)h(not)i(only)e(natural,)h(but)448 4942 y(also)44 b(sho)m(ws)g(that)h(the)f(algebraic)g(metho)s(ds)f(w)m(ork)h(without)f (di\016cult)m(y)g(in)f(a)j(v)m(ery)1920 5225 y(8)p eop %%Page: 9 9 9 8 bop 448 573 a Fy(general)33 b(con)m(text.)50 b(The)32 b(basic)g(examples)h(one)g(should)d(ha)m(v)m(e)k(in)e(mind)e(are)k Ft(X)i Fy(=)29 b Fp(R)3384 540 y FF(n)448 686 y Fy(or)h Fp(Z)624 653 y FF(n)667 686 y Fy(;)g(ho)m(w)m(ev)m(er,)h(the)f(case)h (of)e(\014nite)g(dimensional)e(v)m(ector)k(spaces)f(o)m(v)m(er)h(lo)s (cal)e(\014elds)448 799 y(\(e.g.)53 b Ft(p)p Fy(-adic)34 b(n)m(um)m(b)s(ers\))f(is)g(v)m(ery)h(in)m(teresting)g(and)f(also)h(of) g(some)h(imp)s(ortance)e(\(see)448 912 y([T)-8 b(ai,)31 b(Sa-C]\).)42 b(Man)m(y)31 b(other)g(non)m(trivial)e(examples)h(can)h (b)s(e)e(giv)m(en,)i(lik)m(e)f(the)h(\\)p Ft(p)p Fy(-adic)448 1024 y(torus")39 b(\(the)g(dual)e(of)i(the)g(compact,)j(totally)c (disconnected,)j(non-discrete)d(group)448 1137 y(of)32 b Ft(p)p Fy(-adic)g(in)m(tegers\);)h(see)g([F)-8 b(ol2)q(,)32 b(Gu,)g(W)-8 b(ei)q(].)45 b(But)32 b(w)m(e)h(stress)e(the)h(fact)h (that)g(ev)m(en)f(in)448 1250 y(the)i(simplest)f(situations)f(\()p Ft(X)39 b Fy(=)31 b Fp(R)43 b Fy(or)34 b Fp(Z)p Fy(\))c(the)k (algebraic)g(tec)m(hniques)g(giv)m(e,)h(rather)448 1363 y(easily)-8 b(,)31 b(results)e(whic)m(h)g(do)h(not)h(seem)f(to)h(b)s(e) f(co)m(v)m(ered)i(b)m(y)e(other)h(means.)589 1476 y(In)f(Section)g(2)h (w)m(e)g(discuss)e(sev)m(eral)h(questions)g(concerning)g(the)g (self-adjoin)m(t)g(op-)448 1589 y(erators)k(a\016liated)f(to)h Ft(C)1318 1556 y FG(\003)1356 1589 y Fy(-algebras,)h(their)d(essen)m (tial)h(sp)s(ectra)g(and)g(the)g(connection)448 1702 y(with)42 b(the)h(problem)e(of)j(computing)e(quotien)m(ts)h(of)g Ft(C)2377 1669 y FG(\003)2416 1702 y Fy(-algebras)g(with)e(resp)s(ect)i (to)448 1815 y(some)31 b(ideals.)39 b(Theorem)31 b(2.2)g(will)d(b)s(e)h (particularly)f(useful)h(in)g(a)i(later)f(section.)589 1928 y(Section)43 b(3)h(is)e(dev)m(oted)i(to)g(a)f(short)g(presen)m (tation)g(of)g(the)h(theory)f(of)g(crossed)448 2041 y(pro)s(ducts)24 b(of)i Ft(C)991 2008 y FG(\003)1030 2041 y Fy(-algebras)f(b)m(y)g (actions)h(of)f(ab)s(elian)f(groups)h(with)f(emphasis)f(on)j(some)448 2154 y(results)39 b(that)h(w)m(e)g(need)f(and)g(whic)m(h)f(w)m(e)i(ha)m (v)m(e)h(not)e(b)s(een)g(able)g(to)h(\014nd)e(in)g(the)i(lit-)448 2267 y(erature)g(\(at)g(least)f(in)f(a)h(su\016cien)m(tly)f(explicit)f (form\).)66 b(Esp)s(ecially)37 b(imp)s(ortan)m(t)h(for)448 2379 y(us)g(are)g(Theorem)f(3.3)j(and)d(Theorem)g(3.12;)44 b(for)38 b(the)g(pro)s(of)f(of)h(this)f(last)h(theorem)448 2492 y(w)m(e)h(b)s(ene\014ted)e(from)g(a)h(discussion)e(with)g(G.)j(Sk) -5 b(andalis.)61 b(In)37 b(writing)f(this)h(section)448 2605 y(\(and)i(also)f Fl(x)p Fy(6.1,)k(where)c(sev)m(eral)h(standard)f (de\014nitions)e(and)i(results)f(on)h(en)m(v)m(elop-)448 2718 y(ing)h Ft(C)681 2685 y FG(\003)720 2718 y Fy(-algebras)g(are)h (gathered\),)j(w)m(e)d(had)e(in)h(mind)e(mainly)g(p)s(eople)i(w)m (orking)f(in)448 2831 y(sp)s(ectral)d(theory)h(whic)m(h)e(are)i(not)g (familiar)d(with)h(suc)m(h)i(topics.)56 b(W)-8 b(e)36 b(men)m(tion)f(that)448 2944 y(the)40 b(v)m(ersion)e(of)i(the)f (Riesz-Kolmogoro)m(v)h(compacit)m(y)g(criterion)e(stated)i(in)e Fl(x)p Fy(3.6)j(is)448 3057 y(imp)s(ortan)m(t)35 b(not)h(only)f(for)h 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b(time)f(explicit)f(and)h(highly)f(non)m(trivial.)45 b(One)32 b(of)h(the)f(main)g(tec)m(hnical)g(p)s(oin)m(ts)448 4976 y(of)27 b(our)f(pap)s(er)g(is)g(Theorem)g(5.11)i(\(and)f(its)f (corollary\))h(whic)m(h)e(giv)m(es)i(a)g(description)e(of)1920 5225 y(9)p eop %%Page: 10 10 10 9 bop 448 573 a Fy(this)33 b(algebra)h(in)e(the)i(spirit)d(of)j(the) g(Riesz-Kolmogoro)m(v)h(compacit)m(y)f(criterion)f(\(i.e.)448 686 y(in)c(terms)i(of)f(phase)g(space)h(prop)s(erties)e(of)h(the)h(op)s (erators\).)448 877 y Fz(1.7.)119 b Fy(W)-8 b(e)37 b(recall)f(sev)m (eral)h(notations)g(and)f(con)m(v)m(en)m(tions)h(whic)m(h)f(are)h (usual)e(in)g(the)448 990 y(theory)40 b(of)g Ft(C)928 957 y FG(\003)967 990 y Fy(-algebras.)69 b(A)39 b Fl(\003)p Fy(-homomorphism)g(b)s(et)m(w)m(een)h(t)m(w)m(o)h Ft(C)2863 957 y FG(\003)2902 990 y Fy(-algebras)f(will)448 1103 y(b)s(e)h(called)g Fo(morphism)7 b Fy(.)78 b Fs(A)1467 1078 y Fl(\030)1467 1107 y Fy(=)1582 1103 y Fs(B)41 b Fy(means)h(that)h(the)f Ft(C)2450 1070 y 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2458 y Fy(app)s(ears)g(in)g(a)i(relation)f(in)m(v)m(olving)e(an)i(op)s(erator) h Ft(T)13 b Fy(,)40 b(this)448 2571 y(means)c(that)g(the)f(relation)g (is)f(satis\014ed)h(\(or)h(has)f(to)h(b)s(e)f(satis\014ed\))g(b)s(oth)f (b)m(y)i Ft(T)48 b Fy(and)448 2684 y(b)m(y)31 b Ft(T)641 2651 y FG(\003)680 2684 y Fy(.)589 2796 y(If)26 b Ft(X)33 b Fy(is)25 b(a)h(lo)s(cally)e(compact)j(top)s(ological)e(space)i(then)e Ft(C)2547 2811 y Fq(b)2590 2796 y Fy(\()p Ft(X)7 b Fy(\))27 b(is)e(the)h Ft(C)3080 2763 y FG(\003)3119 2796 y Fy(-algebra)448 2909 y(of)31 b(complex)f(con)m(tin)m(uous)g(b)s(ounded)f(functions)g (on)h Ft(X)7 b Fy(,)31 b Ft(C)2456 2923 y FG(1)2531 2909 y Fy(\()p Ft(X)7 b Fy(\))31 b(the)g Ft(C)2943 2876 y FG(\003)2982 2909 y Fy(-subalgebra)448 3022 y(of)g(functions)d(whic)m (h)h(ha)m(v)m(e)j(a)e(limit)e(at)j(in\014nit)m(y)-8 b(,)29 b(and)g Ft(C)2388 3036 y Fq(0)2428 3022 y Fy(\()p Ft(X)7 b Fy(\))31 b(the)g Ft(C)2840 2989 y FG(\003)2878 3022 y Fy(-subalgebra)f(of)448 3135 y(functions)25 b(con)m(v)m(ergen)m(t)j (to)f(zero)f(at)h(in\014nit)m(y)-8 b(.)37 b(If)26 b(the)g(top)s(ology)g (of)g Ft(X)33 b Fy(is)25 b(asso)s(ciated)h(to)448 3248 y(a)h(giv)m(en)f(uniform)d(structure)j(\(e.g.)h(if)e Ft(X)33 b Fy(is)25 b(a)i(lo)s(cally)d(compact)j(ab)s(elian)e(group\))g (then)448 3361 y Ft(C)520 3328 y Fq(u)513 3389 y(b)563 3361 y Fy(\()p Ft(X)7 b Fy(\))32 b(denotes)e(the)h Ft(C)1304 3328 y FG(\003)1343 3361 y Fy(-algebra)g(of)f(b)s(ounded)e(uniformly)g (con)m(tin)m(uous)i(functions.)589 3474 y(The)39 b(notion)g(of)g (\014lter)g(app)s(ears)f(to)i(b)s(e)f(quite)g(useful)e(for)i(our)g (purp)s(oses.)66 b(The)448 3587 y(reader)45 b(has)f(to)h(kno)m(w)f (only)g(the)h(simplest)d(facts)j(concerning)f(their)g(theory)g(\(the) 448 3700 y(deep)s(est)25 b(thing)e(w)m(e)i(use)g(is)e(that)j(an)e (ultra\014lter)f(on)h(a)h(compact)h(space)f(is)f(con)m(v)m(ergen)m (t\);)448 3813 y(see)39 b([Bou2)q(])f(for)f(a)h(v)m(ery)h(clear)e (presen)m(tation.)64 b(If)37 b Fl(F)47 b Fy(is)37 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Fr(\033)l Fy(\()p Ft(H)7 b Fy(\).)448 686 y(This)23 b(implies)e Ft(')p Fy(\()p Ft(H)7 b Fy(\))26 b Fl(2)f Fs(C)f Fy(for)g(all)f Ft(')j Fl(2)f Ft(C)1850 700 y Fq(0)1889 686 y Fy(\()p Fp(R)s Fy(\))q(,)31 b(so)24 b(eac)m(h)i(self-adjoin)m(t)d(op)s(erator)i(on)e Fn(H)448 799 y Fy(a\016liated)35 b(to)h Fs(C)f Fy(de\014nes)f(an)h (observ)-5 b(able)34 b(a\016liated)g(to)i Fs(C)p Fy(.)55 b(The)35 b(other)g(observ)-5 b(ables)448 912 y(a\016liated)30 b(to)h Fs(C)f Fy(can)h(b)s(e)f(realized)f(as)i Fo(non-densely)g Fy(de\014ned)f(op)s(erators)g(on)g Fn(H)d Fy(.)589 1024 y(If)41 b Ft(H)48 b Fy(is)40 b(a)h(self-adjoin)m(t)g(op)s(erator)g(on)g Fn(H)68 b Fy(a\016liated)40 b(to)i Fs(C)f Fy(and)f(if)g(the)h(corre-) 448 1137 y(sp)s(onding)31 b(observ)-5 b(able)33 b(is)g Fs(C)p Fy(-nondegenerate)h(w)m(e)g(sa)m(y)h(that)f Ft(H)42 b Fo(is)36 b(strictly)g(a\016liate)-5 b(d)448 1250 y(to)52 b Fs(C)p Fy(.)85 b(In)44 b(this)g(case,)50 b(the)45 b(Cohen-Hewitt)g (theorem)h(\(see)g(V.9.2)h(in)c([F)-8 b(eD)r(]\))46 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b(the)h(linear)e(subspace)h (generated)448 3057 y(b)m(y)37 b(the)h(elemen)m(ts)f(of)h(the)f(form)g Ft(S)5 b(f)10 b Fy(,)38 b(with)e Ft(S)42 b Fl(2)36 b Fs(C)i Fy(and)e Ft(f)46 b Fl(2)37 b Fn(H)26 b Fy(,)39 b(is)e(dense)f(in)g Fn(H)p Fy(.)448 3170 y(It)d(can)g(b)s(e)f(sho)m(wn) h(that)g Fo(if)h Fs(C)h Fo(is)g(nonde)-5 b(gener)g(ate)37 b(on)e Fn(H)p Fo(,)f(then)h(the)h(c)-5 b(orr)g(esp)g(ondenc)g(e)448 3283 y(b)g(etwe)g(en)35 b(self-adjoint)h(op)-5 b(er)g(ators)38 b(on)d Fn(H)61 b Fo(strictly)35 b(a\016liate)-5 b(d)36 b(to)f Fs(C)g Fo(and)g(observables)448 3396 y(strictly)f(a\016liate)-5 b(d)34 b(to)g Fs(C)e Fo(de\014ne)-5 b(d)34 b(ab)-5 b(ove)33 b(is)g(bije)-5 b(ctive)p Fy(.)589 3509 y(W)d(e)23 b(stress)e(the)h (fact)g(that)g(if)e Fs(J)i Fy(is)e(an)h(ideal)g(in)f Fs(C)h Fy(and)g Ft(H)28 b Fy(is)20 b(a)i(self-adjoin)m(t)f(op)s(erator) 448 3621 y(a\016liated)37 b(to)h Fs(C)p Fy(,)i(then)d(the)g(quotien)m (t)1819 3598 y Fm(c)1816 3621 y Ft(H)59 b Fy(is)37 b(a)g(w)m(ell)g 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Fy(\))37 b Fl(\032)g Fs(C)g Fy(and)g(let)g(us)g(tak)m(e)i(ab)s(o)m(v) m(e)f Fs(J)f Fy(=)f Ft(K)7 b Fy(\()p Fn(H)27 b Fy(\).)62 b(W)-8 b(e)448 4542 y(recall)33 b(that)i(a)f(real)g(n)m(um)m(b)s(er)e Ft(\025)i Fy(do)s(es)g(not)g(b)s(elong)f(to)h(the)g(essen)m(tial)g(sp)s (ectrum)f(of)h(a)448 4655 y(self-adjoin)m(t)i(op)s(erator)g Ft(H)43 b Fy(if)35 b(and)h(only)f(if)g Ft(')p Fy(\()p Ft(H)7 b Fy(\))36 b Fl(2)e Ft(K)7 b Fy(\()p Fn(H)27 b Fy(\))36 b(for)g(some)g Ft(')g Fl(2)e Ft(C)3261 4669 y Fq(0)3300 4655 y Fy(\()p Fp(R)t Fy(\))448 4777 y(suc)m(h)c(that)h Ft(')p Fy(\()p Ft(\025)p Fy(\))c Fl(6)p Fy(=)e(0.)41 b(Hence)31 b(if)e Ft(H)38 b Fy(is)29 b(a\016liated)h(to)h Fs(C)f Fy(w)m(e)h(get)h Fr(\033)2740 4791 y Fq(ess)2831 4777 y Fy(\()p Ft(H)7 b Fy(\))26 b(=)f Fr(\033)l Fy(\()3203 4754 y Fm(c)3199 4777 y Ft(H)d Fy(\).)448 4968 y Fz(2.3.)85 b Fy(The)30 b(algebras)h(that)h(w)m(e)f(consider)f(ha)m(v)m(e)i(to)g(b) s(e)e(rather)h(small,)f(suc)m(h)g(that)i(the)1897 5225 y(12)p eop %%Page: 13 13 13 12 bop 448 573 a Fy(quotien)m(t)30 b(with)d(resp)s(ect)j(to)f(the)h (ideal)e(of)h(compact)h(op)s(erators)g(b)s(e)e(computable.)40 b(On)448 686 y(the)24 b(other)g(hand)e(w)m(e)i(w)m(ould)f(lik)m(e)f (that)i(the)g(class)f(of)h(self-adjoin)m(t)f(op)s(erators)h (a\016liated)448 799 y(to)k(it)f(b)s(e)f(large.)40 b(So)27 b(w)m(e)h(are)f(in)m(terested)g(in)f(ha)m(ving)h(e\016cien)m(t)g (a\016liation)f(criteria.)39 b(W)-8 b(e)448 912 y(recall)30 b(t)m(w)m(o)i(suc)m(h)e(criteria)f(b)s(elo)m(w.)589 1024 y(W)-8 b(e)43 b(remain)e(in)g(the)h(setting)f(of)h Fl(x)p Fy(2.2)i(and)d(consider)g(a)h(self-adjoin)m(t)f(op)s(erator)448 1137 y Ft(H)524 1151 y Fq(0)603 1137 y Fy(on)f Fn(H)26 b Fy(.)69 b(W)-8 b(e)41 b(sa)m(y)g(that)f Ft(V)60 b Fy(is)39 b(a)h Fo(standar)-5 b(d)44 b(form)e(p)-5 b(erturb)g(ation)49 b Fy(of)40 b Ft(H)3107 1151 y Fq(0)3186 1137 y Fy(if)f Ft(V)60 b Fy(a)448 1250 y(con)m(tin)m(uous)35 b(symmetric)f (sesquilinear)e(form)j(on)f Fn(G)49 b Fy(=)32 b Ft(D)s Fy(\()p Fl(j)p Ft(H)2620 1264 y 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Fy([)p Ft(T)e Fy(\()p Ft(x)21 b Fl(\000)f Ft(y)s Fy(\)])30 b(d)p Ft(y)s(;)-2288 b Fy(\(3.12\))1287 3850 y Ft(S)1348 3812 y Fl(\003)1397 3850 y Fy(\()p Ft(x)p Fy(\))84 b(=)f Ft(\013)1815 3864 y FF(x)1859 3850 y Fy([)p Ft(S)5 b Fy(\()p Fl(\000)p Ft(x)p Fy(\))2138 3812 y Fl(\003)2188 3850 y Fy(])p Ft(:)-1790 b Fy(\(3.13\))504 4054 y(Note)29 b(that)g Ft(C)981 4068 y Fq(c)1016 4054 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)22 b Fy(\),)29 b(the)f(space)h(of)f(con)m(tin)m(uous)f(functions)g Ft(X)32 b Fl(!)26 b Fn(A)49 b Fy(with)26 b(com-)448 4167 y(pact)k(supp)s(ort,)d(is)h(a)h(dense)f Fl(\003)p Fy(-subalgebra)h(of)g Ft(L)2113 4134 y Fq(1)2152 4167 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)23 b Fy(\).)40 b(Moreo)m(v)m(er,)31 b(the)e(algebraic)448 4280 y(tensor)h(pro)s(duct)e Fn(A)39 b Fl(\012)1244 4295 y Fq(alg)1357 4280 y Ft(C)1422 4294 y Fq(c)1457 4280 y Fy(\()p Ft(X)7 b Fy(\))31 b(is)d(a)i(dense)f(subspace)g(\(iden)m (ti\014ed)e(with)h(the)i(set)g(of)448 4393 y(elemen)m(ts)38 b(from)e Ft(C)1113 4407 y Fq(c)1149 4393 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)22 b Fy(\))37 b(whose)g(ranges)h(are)f(con)m(tained)g (in)f(\014nite)g(dimensional)448 4506 y(subspaces)30 b(of)h Fn(A)21 b Fy(\).)589 4619 y(Assume,)48 b(furthermore,)f(that)e Fn(A)65 b Fy(is)43 b(realized)h(on)g(a)h(Hilb)s(ert)d(space)j Fn(H)71 b Fy(and)448 4731 y(let)35 b Fn(H)673 4745 y FF(X)771 4731 y Fy(=)d Ft(L)936 4698 y Fq(2)975 4731 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(H)28 b Fy(\).)53 b(Then)33 b(one)i(has)f(a)h(faithful)d(represen)m(tation)j(of)f Ft(L)3110 4698 y Fq(1)3149 4731 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)23 b Fy(\))448 4844 y(on)36 b Fn(H)669 4858 y FF(X)736 4844 y Fy(,)i(the)e(so-called)f Fo(left)j(r)-5 b(e)g(gular)39 b(r)-5 b(epr)g(esentation)7 b Fy(:)55 b(one)36 b(de\014nes)f(the)i(action)f(of)1897 5225 y(17)p eop %%Page: 18 18 18 17 bop 448 573 a Ft(S)31 b Fl(2)24 b Ft(L)682 540 y Fq(1)722 573 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)22 b Fy(\))31 b(on)m(to)g Ft(\030)f Fl(2)24 b Fn(H)1488 587 y FF(X)1586 573 y Fy(b)m(y)448 814 y(\(3.14\))512 b(\()p Ft(S)26 b Fl(\017)21 b Ft(\030)t Fy(\)\()p Ft(x)p Fy(\))26 b(=)1696 691 y Fm(Z)1746 897 y FF(X)1829 814 y Ft(\013)1887 828 y FG(\000)p FF(x)1986 814 y Fy([)p Ft(S)5 b Fy(\()p Ft(x)20 b Fl(\000)g Ft(y)s Fy(\)])15 b Ft(\030)t Fy(\()p Ft(y)s Fy(\))g(d)q Ft(y)s(:)448 1068 y Fy(In)30 b(particular,)f Ft(L)1069 1035 y Fq(1)1108 1068 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)23 b Fy(\))33 b Fo(is)f(an)h Ft(A)1724 1035 y FG(\003)1764 1068 y Fo(-algebr)-5 b(a)38 b Fy(\(see)31 b Fl(x)p Fy(6.1)h(in)d(the)i(App)s(endix\).)448 1252 y Fz(De\014nition)36 b(3.1)46 b Fo(If)34 b Fn(A)56 b Fo(is)34 b(a)h Ft(X)7 b Fo(-algebr)-5 b(a,)37 b(then)e(the)g Fr(cr)s(ossed)e(pr)s(oduct)h Fo(of)h Fn(A)56 b Fo(by)448 1364 y(the)29 b(action)h Ft(\013)f Fo(of)f Ft(X)7 b Fo(,)30 b(is)e(the)h(enveloping)g Ft(C)1951 1331 y FG(\003)1990 1364 y Fo(-algebr)-5 b(a)30 b(of)e(the)h Ft(A)2644 1331 y FG(\003)2684 1364 y Fo(-algebr)-5 b(a)29 b Ft(L)3082 1331 y Fq(1)3121 1364 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)23 b Fy(\))p Fo(.)448 1477 y(This)33 b Ft(C)729 1444 y FG(\003)768 1477 y Fo(-algebr)-5 b(a)34 b(is)e(denote)-5 b(d)34 b(by)f Fn(A)41 b Fj(o)20 b Ft(X)7 b Fo(.)448 1661 y Fy(Th)m(us)24 b Fn(A)29 b Fj(o)8 b Ft(X)32 b Fy(is)24 b(the)h(completion)e(of)i Ft(L)1816 1628 y Fq(1)1855 1661 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)23 b Fy(\))i(under)d(the)j(largest)g Ft(C)2930 1628 y FG(\003)2969 1661 y Fy(-norm)f(on)g(it,)448 1774 y(and)29 b(eac)m(h)i(represen)m(tation)e(of)h Ft(L)1586 1741 y Fq(1)1625 1774 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)22 b Fy(\))30 b(extends)f(to)i(a)e(represen)m(tation)h(of)f Fn(A)40 b Fj(o)18 b Ft(X)448 1887 y Fy(\(see)38 b(the)f Fl(x)p Fy(6.1)h(in)d(the)i(App)s(endix\).)58 b(Due)37 b(to)g(the)g(fact)g(that)h Ft(X)44 b Fy(is)36 b(ab)s(elian)f(\(hence) 448 2000 y(amenable\))40 b(the)f(crossed)g(pro)s(duct)f(de\014ned)g(ab) s(o)m(v)m(e)j(coincides)d(with)g(the)h(so-called)448 2113 y(\\reduced)30 b(crossed)h(pro)s(duct")e(\(cf.)j(Theorems)d(7.7.5) k(and)c(7.7.7)k(in)c([P)m(ed]\):)448 2297 y Fz(Theorem)34 b(3.2)46 b Fo(The)i(left)g(r)-5 b(e)g(gular)50 b(r)-5 b(epr)g(esentation)50 b(of)f Ft(L)2553 2264 y Fq(1)2592 2297 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)22 b Fy(\))49 b Fo(extends)f(to)h(a)448 2410 y(faithful)35 b(r)-5 b(epr)g(esentation) 38 b(of)c Fn(A)43 b Fj(o)21 b Ft(X)7 b Fo(.)47 b(In)35 b(p)-5 b(articular,)37 b Fn(A)42 b Fj(o)22 b Ft(X)41 b Fo(is)35 b(c)-5 b(anonic)g(al)5 b(ly)36 b(iso-)448 2522 y(morphic)h(to)f(the)g(closur)-5 b(e)36 b(in)f Ft(B)5 b Fy(\()p Fn(H)1691 2536 y FF(X)1758 2522 y Fy(\))35 b Fo(of)h(the)f Fl(\003)p Fo(-algebr)-5 b(a)37 b(of)e(op)-5 b(er)g(ators)39 b(of)c(the)h(form)448 2635 y(\(3.14\).)448 2819 y Fy(Heuristically)-8 b(,)25 b(one)h(should)e(think)h(of)h Fn(A)32 b Fj(o)11 b Ft(X)33 b Fy(as)26 b(a)g(kind)e(of)i(t)m(wisted)f (tensor)h(pro)s(duct)448 2932 y(of)j(algebras)f Fn(A)49 b Fy(and)28 b Ft(C)1264 2946 y Fq(0)1303 2932 y Fy(\()p Ft(X)1420 2899 y FG(\003)1460 2932 y Fy(\),)h(where)f Ft(X)1892 2899 y FG(\003)1960 2932 y Fy(is)f(the)i(group)e(dual)g(to)i Ft(X)7 b Fy(.)40 b(In)28 b(fact,)i(if)d(the)448 3045 y(action)k(of)g Ft(X)37 b Fy(on)31 b Fn(A)51 b Fy(is)30 b(trivial,)e(then)i Fn(A)42 b Fj(o)20 b Ft(X)32 b Fy(=)25 b Fn(A)42 b Fl(\012)20 b Ft(C)2462 3059 y Fq(0)2501 3045 y Fy(\()p Ft(X)2618 3012 y FG(\003)2658 3045 y Fy(\).)448 3236 y Fz(3.2.)141 b Fy(The)40 b(corresp)s(ondence)f Fn(A)63 b Fl(7!)42 b Fn(A)48 b Fj(o)27 b Ft(X)47 b Fy(extends)40 b(to)h(a)g(co)m(v)-5 b(arian)m(t)42 b(functor)448 3349 y(from)g(the)g(category)h(of)f Ft(X)7 b Fy(-algebras)43 b(\(with)d Ft(X)7 b Fy(-morphisms)40 b(as)i(morphisms\))e(in)m(to)448 3462 y(the)33 b(category)h(of)f Ft(C)1153 3429 y FG(\003)1192 3462 y Fy(-algebras.)47 b(Indeed,)32 b(if)f Ft(\036)e Fy(:)g Fn(A)49 b Fl(!)29 b Fn(B)36 b Fy(is)31 b(a)i Ft(X)7 b Fy(-morphism,)31 b(then)448 3575 y(it)36 b(clearly)f(induces)g(a)i (morphism)c Ft(\036)1737 3589 y Fq(0)1812 3575 y Fy(:)i Ft(L)1934 3542 y Fq(1)1973 3575 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)23 b Fy(\))35 b Fl(!)g Ft(L)2484 3542 y Fq(1)2523 3575 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(B)t Fy(\))37 b(b)m(y)f(the)h(form) m(ula)448 3688 y(\()p Ft(\036)537 3702 y Fq(0)577 3688 y Ft(S)5 b Fy(\)\()p Ft(x)p Fy(\))36 b(:=)f Ft(\036)p Fy([)p Ft(S)5 b Fy(\()p Ft(x)p Fy(\)].)59 b(Hence)37 b(w)m(e)g(ma)m(y)g(de\014ne)e(the)h(morphism)e Ft(\036)2867 3721 y Fl(\003)2956 3688 y Fy(:)h Fn(A)45 b Fj(o)24 b Ft(X)42 b Fl(!)448 3801 y Fn(B)33 b Fj(o)c Ft(X)52 b Fy(as)44 b(the)g(canonical)g(extension)f(of)h Ft(\036)2085 3815 y Fq(0)2169 3801 y Fy(to)h(the)f(en)m(v)m(eloping)f(algebras,)48 b(i.e.)448 3914 y Ft(\036)502 3947 y Fl(\003)581 3914 y Fy(=)25 b(\()p Ft(\036)766 3928 y Fq(0)806 3914 y Fy(\))841 3947 y Fl(\003)896 3914 y Fy(.)448 4097 y Fz(Theorem)34 b(3.3)46 b Fo(L)-5 b(et)33 b Fn(J)18 b Fo(,)32 b Fn(A)21 b Fo(,)33 b Fn(B)j Fo(b)-5 b(e)32 b Ft(X)7 b Fo(-algebr)-5 b(as)34 b(and)g(let)1066 4351 y Fy(0)47 b Fl(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)47 b Fn(J)1711 4290 y Ft(\036)1614 4351 y Fl(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)46 b Fn(A)2143 4290 y Ft( )2050 4351 y Fl(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)46 b Fn(B)k Fl(\000)-42 b(\000)-20 b(\000)f(\000)-42 b(!)46 b Fy(0)448 4535 y Fo(b)-5 b(e)33 b(an)g(exact)g(se)-5 b(quenc)g(e)32 b(of)h Ft(X)7 b Fo(-morphisms.)44 b(Then)776 4796 y Fy(0)j Fl(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)46 b Fn(J)38 b Fj(o)20 b Ft(X)1587 4720 y(\036)1641 4753 y Fl(\003)1517 4796 y(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)46 b Fn(A)c Fj(o)20 b Ft(X)2215 4720 y( )2274 4753 y Fl(\003)2147 4796 y(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)46 b Fn(B)24 b Fj(o)c Ft(X)54 b Fl(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)46 b Fy(0)448 4976 y Fo(is)33 b(an)g(exact)g(se)-5 b(quenc)g(e.)1897 5225 y Fy(18)p eop %%Page: 19 19 19 18 bop 448 573 a Fz(Pro)s(of:)48 b Fy(It)30 b(su\016ces)g(to)h(pro)m (v)m(e)h(that)628 785 y(0)46 b Fl(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)45 b Ft(L)1075 752 y Fq(1)1115 785 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(J)j Fy(\))1545 724 y Ft(\036)1599 738 y Fq(0)1467 785 y Fl(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)46 b Ft(L)1824 752 y Fq(1)1863 785 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)22 b Fy(\))2271 724 y Ft( )2330 738 y Fq(0)2196 785 y Fl(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)46 b Ft(L)2553 752 y Fq(1)2592 785 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(B)t Fy(\))47 b Fl(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)45 b Fy(0)448 935 y(is)33 b(an)h(exact)h(sequence)g(of)f(Banac)m(h)h Fl(\003)p Fy(-algebras;)h(then)e(w)m(e)g(use)g(Theorem)f(6.1.)53 b(The)448 1048 y(injectivit)m(y)31 b(of)g Ft(\036)1037 1062 y Fq(0)1108 1048 y Fy(and)g(the)h(relation)f Ft( )1838 1062 y Fq(0)1898 1048 y Fl(\016)22 b Ft(\036)2019 1062 y Fq(0)2085 1048 y Fy(=)27 b(0)32 b(are)g(ob)m(vious.)43 b(If)31 b Ft(S)h Fl(2)27 b Ft(L)3110 1015 y Fq(1)3149 1048 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)23 b Fy(\))448 1161 y(and)j Ft( )680 1175 y Fq(0)719 1161 y Fy(\()p Ft(S)5 b Fy(\))26 b(=)f(0)i(then)f Ft( )s Fy([)p Ft(S)5 b Fy(\()p Ft(x)p Fy(\)])26 b(=)f(0)i(for)f(a.e.)h Ft(x)e Fl(2)g Ft(X)7 b Fy(,)28 b(i.e.)e Ft(S)5 b Fy(\()p Ft(X)i Fy(\))26 b Fl(2)f Ft(\036)p Fy(\()p Fn(J)18 b Fy(\))27 b(for)f(a.e.)h Ft(x)p Fy(.)448 1274 y(But)33 b Ft(\036)f Fy(is)g(an)g(isometry)-8 b(,)33 b(so)f(there)h(is)e Ft(T)42 b Fl(2)28 b Ft(L)2018 1241 y Fq(1)2057 1274 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(J)k Fy(\))32 b(suc)m(h)g(that)h Ft(S)5 b Fy(\()p Ft(X)i Fy(\))30 b(=)e Ft(\036)p Fy([)p Ft(T)13 b Fy(\()p Ft(x)p Fy(\)])448 1387 y(for)30 b(a.e.)i Ft(x)p Fy(.)41 b(This)28 b(pro)m(v)m(es)j(that)g(k)m(er)16 b Ft( )1753 1401 y Fq(0)1818 1387 y Fy(=)25 b Ft(\036)1968 1401 y Fq(0)2007 1387 y Fy(\()p Ft(L)2104 1354 y Fq(1)2144 1387 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(J)k Fy(\)\).)589 1500 y(The)28 b(surjectivit)m(y)g(of)h Ft(\036)1404 1514 y Fq(0)1472 1500 y Fy(is)e(a)i(consequence)g(of)g(the)f(follo)m(wing)f (general)i(prop)s(ert)m(y)448 1612 y(\(see)k Fl(x)p Fy(3.5)g(in)d ([DeFl)q(]\).)44 b(Let)32 b Fg(E)f Fy(b)s(e)g(a)h(Banac)m(h)g(space)h (and)d Ft(F)41 b Fl(2)26 b Ft(L)2728 1579 y Fq(1)2767 1612 y Fy(\()p Ft(X)7 b Fy(;)15 b Fg(E)p Fy(\).)46 b(Then)30 b(for)448 1725 y(eac)m(h)36 b(n)m(um)m(b)s(er)e Ft(")f(>)g Fy(0)i(there)g(are)g(sequences)g Ft(f)2107 1739 y FF(n)2187 1725 y Fl(2)d Ft(L)2342 1692 y Fq(1)2381 1725 y Fy(\()p Ft(X)7 b Fy(\))36 b(and)f Ft(e)2793 1739 y FF(n)2873 1725 y Fl(2)d Fg(E)j Fy(suc)m(h)f(that)448 1838 y Ft(F)39 b Fy(=)641 1770 y Fm(P)737 1796 y FG(1)737 1865 y FF(n)p Fq(=1)889 1838 y Ft(e)931 1852 y FF(n)998 1838 y Fl(\012)20 b Ft(f)1134 1852 y FF(n)1211 1838 y Fy(and)1088 1964 y FG(1)1058 1991 y Fm(X)1057 2186 y FF(n)p Fq(=1)1205 2078 y Fl(k)p Ft(e)1292 2092 y FF(n)1340 2078 y Fl(k)1400 1954 y Fm(Z)1507 2078 y Fl(j)p Ft(f)1577 2092 y FF(n)1623 2078 y Fy(\()p Ft(x)p Fy(\))p Fl(j)15 b Fy(d)q Ft(x)25 b Fl(\024)g Fy(\(1)c(+)f Ft(")p Fy(\))2294 1954 y Fm(Z)2401 2078 y Fl(k)p Ft(F)13 b Fy(\()p Ft(x)p Fy(\))p Fl(k)i Fy(d)q Ft(x:)448 2324 y Fy(Note)36 b(also)e(that)h(since)f(the)h(map)f Fn(A)21 b Ft(=)15 b Fy(k)m(er)h Ft( )36 b Fl(!)c Fn(B)38 b Fy(induced)32 b(b)m(y)i Ft( )k Fy(is)33 b(an)i(isometric)448 2437 y(bijection,)40 b(for)e(eac)m(h)i Ft(b)f Fl(2)g Fn(B)j Fy(there)d(is)e Ft(a)i Fl(2)g Fn(A)60 b Fy(suc)m(h)38 b(that)h Ft( )s Fy(\()p Ft(a)p Fy(\))h(=)f Ft(b)f Fy(and)g Fl(k)p Ft(a)p Fl(k)i Fy(=)448 2550 y(\(1)21 b(+)f Ft(")p Fy(\))p Fl(k)p Ft(b)p Fl(k)p Fy(.)p 3371 2542 67 67 v 589 2700 a(Let)27 b Fn(J)43 b Fy(b)s(e)25 b(a)h(stable)f(ideal)g(of)h (a)g Ft(X)7 b Fy(-algebra)26 b Fn(A)c Fy(.)39 b(According)25 b(to)i(Theorem)e(3.3,)j(if)448 2813 y Ft(j)j Fy(:)26 b Fn(J)43 b Fl(!)25 b Fn(A)47 b Fy(is)26 b(the)g(inclusion)e(map,)j (then)f Ft(j)2022 2827 y FG(\003)2087 2813 y Fy(:)f Fn(J)30 b Fj(o)12 b Ft(X)33 b Fl(!)25 b Fn(A)33 b Fj(o)12 b Ft(X)34 b Fy(is)25 b(an)i(isometric)448 2926 y(morphism)d(of)j Fn(J)i Fj(o)12 b Ft(X)34 b Fy(on)m(to)27 b(an)f(ideal)f(of)i Fn(A)33 b Fj(o)12 b Ft(X)7 b Fy(.)40 b(F)-8 b(rom)26 b(no)m(w)h(on)f Fo(we)j(shal)5 b(l)30 b(identify)448 3039 y Fn(J)35 b Fj(o)17 b Ft(X)39 b Fo(with)32 b(its)f(image)h(under)f Ft(j)1661 3053 y FG(\003)1701 3039 y Fy(.)40 b(More)30 b(explicitly)-8 b(,)27 b Fn(J)35 b Fj(o)17 b Ft(X)36 b Fy(is)28 b(just)h(the)g(closure)448 3152 y(in)g Fn(A)42 b Fj(o)20 b Ft(X)37 b Fy(of)31 b(the)g(ideal)e Ft(L)1412 3119 y Fq(1)1451 3152 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(J)k Fy(\))30 b(of)h Ft(L)1954 3119 y Fq(1)1993 3152 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)23 b Fy(\).)589 3265 y(No)m(w)32 b(the)g(quotien)m(t)f Ft(C)1386 3232 y FG(\003)1425 3265 y Fy(-algebra)h Fn(B)e Fy(=)d Fn(A)21 b Ft(=)p Fn(J)49 b Fy(has)31 b(a)h(natural)e(structure)h(of)g Ft(X)7 b Fy(-)448 3378 y(algebra)36 b(suc)m(h)g(that)g(the)h(canonical)e (morphism)f Fn(A)56 b Fl(!)34 b Fn(A)21 b Ft(=)p Fn(J)54 b Fy(is)35 b(a)h Ft(X)7 b Fy(-morphism.)448 3490 y(The)44 b(Theorem)g(3.3)h(sa)m(ys)f(also)g(that)h(the)f(morphism)e Fn(A)50 b Fj(o)29 b Ft(X)56 b Fl(!)48 b Fy([)p Fn(A)21 b Ft(=)p Fn(J)d Fy(])30 b Fj(o)f Ft(X)448 3603 y Fy(asso)s(ciated)22 b(to)f(it)g(has)g Fn(J)d Fj(o)q Ft(X)29 b Fy(as)21 b(k)m(ernel.)37 b(W)-8 b(e)22 b(th)m(us)f(get)h(the)f(follo)m(wing)e(reform)m(ulation) 448 3716 y(of)31 b(Theorem)f(3.3:)448 3872 y Fz(Theorem)k(3.4)46 b Fo(If)33 b Fn(J)50 b Fo(is)32 b(a)h(stable)h(ide)-5 b(al)33 b(of)g(a)g Ft(X)7 b Fo(-algebr)-5 b(a)34 b Fn(A)54 b Fo(then)448 4063 y Fy(\(3.15\))1287 4028 y Fn(A)41 b Fj(o)20 b Ft(X)1559 3962 y Fm(.)1603 4106 y Fn(J)38 b Fj(o)20 b Ft(X)1968 4038 y Fl(\030)1968 4067 y Fy(=)2096 4063 y([)q Fn(A)h Ft(=)-15 b Fn(J)18 b Fy(])i Fj(o)g Ft(X)r(:)589 4256 y Fy(The)33 b(simplest)f(case)i(of)g(the)g(preceding) e(situation)g(is)h(that)h(when)e(the)i(exact)h(se-)448 4369 y(quence)27 b(splits,)f(so)h(that)g Fn(A)22 b Ft(=)-15 b Fn(J)44 b Fy(can)27 b(b)s(e)f(realized)g(as)h(a)g(stable)g Ft(C)2697 4336 y FG(\003)2736 4369 y Fy(-subalgebra)f(of)h Fn(A)21 b Fy(.)448 4482 y(Then)30 b(w)m(e)h(ha)m(v)m(e:)448 4638 y Fz(Corollary)k(3.5)46 b Fo(L)-5 b(et)28 b Fn(A)48 b Fo(b)-5 b(e)27 b(a)g Ft(X)7 b Fo(-algebr)-5 b(a,)29 b Fn(J)44 b Fo(a)28 b(stable)f(ide)-5 b(al,)29 b(and)f Fn(B)j Fo(a)c(stable)h Ft(C)3366 4605 y FG(\003)3405 4638 y Fo(-)448 4751 y(sub)-5 b(algebr)g(a)28 b(such)f(that)h Fn(A)46 b Fy(=)25 b Fn(B)11 b Fy(+)c Fn(J)43 b Fo(dir)-5 b(e)g(ct)28 b(line)-5 b(ar)28 b(sum.)39 b(Then)27 b Fn(J)e Fj(o)7 b Ft(X)33 b Fo(is)27 b(an)g(ide)-5 b(al)448 4863 y(in)26 b Fn(A)g Fj(o)5 b Ft(X)i Fo(,)27 b Fn(B)9 b Fj(o)c Ft(X)32 b Fo(is)26 b(a)g Ft(C)1375 4830 y FG(\003)1414 4863 y Fo(-sub)-5 b(algebr)g(a)26 b(of)g Fn(A)g Fj(o)5 b Ft(X)i Fo(,)27 b(and)g Fn(A)f Fj(o)5 b Ft(X)32 b Fy(=)25 b Fn(B)9 b Fj(o)c Ft(X)12 b Fy(+)5 b Fn(J)22 b Fj(o)5 b Ft(X)448 4976 y Fo(dir)-5 b(e)g(ct)34 b(line)-5 b(ar)34 b(sum.)1897 5225 y Fy(19)p eop %%Page: 20 20 20 19 bop 448 573 a Fz(Corollary)35 b(3.6)46 b Fo(L)-5 b(et)40 b Fn(A)21 b Fo(,)41 b Fn(B)i Fo(b)-5 b(e)39 b Ft(X)7 b Fo(-algebr)-5 b(as)40 b(and)g(let)g Fn(A)46 b Fl(\010)25 b Fn(B)42 b Fo(b)-5 b(e)39 b(e)-5 b(quipp)g(e)g(d)41 b(with)448 686 y(the)33 b(natur)-5 b(al)35 b Ft(X)7 b Fo(-algebr)-5 b(a)34 b(structur)-5 b(e.)42 b(Then)448 890 y Fy(\(3.16\))503 b(\()p Fn(A)42 b Fl(\010)20 b Fn(B)t Fy(\))g Fj(o)g Ft(X)1763 865 y Fl(\030)1763 894 y Fy(=)1859 890 y(\()p Fn(A)41 b Fj(o)20 b Ft(X)7 b Fy(\))21 b Fl(\010)f Fy(\()p Fn(B)k Fj(o)c Ft(X)7 b Fy(\))p Ft(:)589 1103 y Fy(W)-8 b(e)32 b(men)m(tion)e(one)h(more)f(fact:)448 1290 y Fz(Prop)s(osition)37 b(3.7)46 b Fo(If)35 b Ft(\036)d Fy(:)f Fn(A)53 b Fl(!)31 b Fn(B)40 b Fo(is)35 b(an)i(inje)-5 b(ctive)35 b(or)i(surje)-5 b(ctive)36 b Ft(X)7 b Fo(-mor)q(phism)448 1403 y(then)35 b Ft(\036)706 1436 y Fl(\003)788 1403 y Fy(:)29 b Fn(A)42 b Fj(o)21 b Ft(X)36 b Fl(!)28 b Fn(B)d Fj(o)c Ft(X)42 b Fo(is)34 b(inje)-5 b(ctive)34 b(or)g(surje)-5 b(ctive)34 b(r)-5 b(esp)g(e)g(ctively.)49 b(In)34 b(p)-5 b(arti-)448 1516 y(cular,)36 b(if)e Fn(A)55 b Fo(is)34 b(a)h(stable)g Ft(C)1435 1483 y FG(\003)1474 1516 y Fo(-sub)-5 b(algebr)g(a)35 b(of)g(the)f Ft(X)7 b Fo(-algebr)-5 b(a)36 b Fn(B)t Fo(,)e(then)h Fn(A)43 b Fj(o)21 b Ft(X)41 b Fo(c)-5 b(an)448 1629 y(b)g(e)33 b(identi\014e)-5 b(d)33 b(with)h(a)f Ft(C)1301 1596 y FG(\003)1340 1629 y Fo(-sub)-5 b(algebr)g(a)33 b(of)g Fn(B)24 b Fj(o)c Ft(X)7 b Fo(.)448 1817 y Fy(The)36 b(assertion)h(is)e(ob)m(vious)h(in)g(the)g(surjectiv)m (e)h(case.)60 b(F)-8 b(or)38 b(the)f(injectiv)m(e)f(case,)j(see)448 1929 y(Prop)s(osition)33 b(7.7.9)i(in)e([P)m(ed)q(].)52 b(So)34 b(what)g(w)m(e)h(pro)m(v)m(ed)f(ab)s(o)m(v)m(e)i(for)e(ideals)f (is)g(v)-5 b(alid)32 b(for)448 2042 y(subalgebras)d(to)s(o.)448 2230 y Fz(Prop)s(osition)37 b(3.8)46 b Fo(L)-5 b(et)24 b Fn(A)45 b Fo(b)-5 b(e)23 b(a)h Ft(X)7 b Fo(-algebr)-5 b(a)25 b(and)g(let)f Fn(B)j Fo(b)-5 b(e)24 b(a)g(nucle)-5 b(ar)25 b(\(e.g.)e(ab)-5 b(elian\))448 2343 y Ft(C)520 2310 y FG(\003)559 2343 y Fo(-algebr)g(a.)40 b(Equip)25 b Fn(A)g Fl(\012)t Fn(B)i Fo(with)f(the)g Ft(X)7 b Fo(-algebr)-5 b(a)26 b(structur)-5 b(e)26 b(de\014ne)-5 b(d)26 b(by)e Ft(\013)3061 2357 y FF(x)3105 2343 y Fy(\()p Ft(a)t Fl(\012)t Ft(b)p Fy(\))h(=)448 2456 y Ft(\013)506 2470 y FF(x)550 2456 y Fy(\()p Ft(a)p Fy(\))c Fl(\012)f Ft(b)p Fo(.)42 b(Then)448 2660 y Fy(\(3.17\))635 b(\()p Fn(A)42 b Fl(\012)20 b Fn(B)t Fy(\))g Fj(o)g Ft(X)1895 2635 y Fl(\030)1895 2664 y Fy(=)1991 2660 y(\()p Fn(A)41 b Fj(o)20 b Ft(X)7 b Fy(\))21 b Fl(\012)f Fn(B)t Ft(:)448 2864 y Fy(Prop)s(osition)29 b(2.4)i(in)e([T)-8 b(ak)q(])31 b(asserts)g(more)f(than)g(this.)40 b(Since)30 b(T)-8 b(ak)j(ai)30 b(giv)m(es)h(no)g(pro)s(of,)448 2977 y(w)m(e)g(shall)e(sk)m(etc)m(h)j(a)e(simple)f(one)h(in)f Fl(x)p Fy(6.2)j(of)f(the)f(App)s(endix.)448 3168 y Fz(3.3.)171 b Fy(W)-8 b(e)46 b(discuss)d(no)m(w)i(the)h(b)s(eha)m(vior)e(of)h(the)h (crossed)f(pro)s(duct)f(under)g(in\014-)448 3281 y(nite)36 b(direct)f(pro)s(ducts)f(and)h(sums.)56 b(Let)36 b Fl(f)p Fn(A)2029 3295 y FF(i)2058 3281 y Fl(g)2103 3295 y FF(i)p FG(2)p FF(I)2250 3281 y Fy(b)s(e)g(an)f(arbitrary)g(family)f(of)i Ft(C)3368 3248 y FG(\003)3407 3281 y Fy(-)448 3394 y(algebras.)49 b(Assume)32 b(that)h(eac)m(h)h Fn(A)1664 3408 y FF(i)1726 3394 y Fy(is)e(a)h Ft(X)7 b Fy(-algebra,)35 b(the)e(corresp)s(onding)e (group)h(of)448 3507 y(automorphisms)h(b)s(eing)g Ft(\013)1397 3474 y FF(i)1426 3507 y Fy(.)53 b(Then)33 b(one)i(ma)m(y)g(de\014ne)e Ft(\013)g Fy(:)f Ft(X)40 b Fl(!)32 b Fy(Aut\()2958 3439 y Fm(Q)3044 3534 y FF(i)p FG(2)p FF(I)3170 3507 y Fn(A)3243 3521 y FF(i)3271 3507 y Fy(\))j(b)m(y)448 3620 y Ft(\013)506 3634 y FF(x)550 3620 y Fy([\()p Ft(a)658 3634 y FF(i)687 3620 y Fy(\))722 3634 y FF(i)p FG(2)p FF(I)834 3620 y Fy(])25 b(=)g(\()p Ft(\013)1073 3587 y FF(i)1073 3642 y(x)1117 3620 y Fy([)p Ft(a)1190 3634 y FF(i)1219 3620 y Fy(]\))1279 3634 y FF(i)p FG(2)p FF(I)1390 3620 y Fy(.)40 b(In)26 b(this)g(w)m(a)m(y)i(w)m(e)g(do)f(not)g(\(in)f(general\))i(get) g(a)f Ft(X)7 b Fy(-algebra)448 3733 y(structure)26 b(on)956 3665 y Fm(Q)1042 3760 y FF(i)p FG(2)p FF(I)1168 3733 y Fn(A)1241 3747 y FF(i)1296 3733 y Fy(b)s(ecause)h(the)f(con)m(tin)m (uit)m(y)g(condition)f(is)h(not)h(satis\014ed.)38 b(Ho)m(w-)448 3846 y(ev)m(er,)50 b(w)m(e)c(ma)m(y)f(de\014ne)g(an)f(\\equicon)m(tin)m (uous)h(pro)s(duct")f(algebra)h(as)g(the)h(largest)448 3959 y(subalgebra)30 b(on)g(whic)m(h)f Ft(\013)i Fy(acts)g(con)m(tin)m (uously:)674 4095 y Fm(Q)760 4121 y FF(X)760 4190 y(i)p FG(2)p FF(I)886 4163 y Fn(A)959 4177 y FF(i)1013 4163 y Fy(=)25 b Fl(f)p Fy(\()p Ft(a)1237 4177 y FF(i)1266 4163 y Fy(\))1301 4177 y FF(i)p FG(2)p FF(I)1438 4163 y Fl(2)1523 4095 y Fm(Q)1609 4190 y FF(i)p FG(2)p FF(I)1735 4163 y Fn(A)1808 4177 y FF(i)1862 4163 y Fl(j)h Fy(lim)2039 4177 y FF(x)p FG(!)p Fq(0)2204 4163 y Fy(sup)2341 4185 y FF(i)p FG(2)p FF(I)2467 4163 y Fl(k)p Ft(\013)2570 4130 y FF(i)2570 4185 y(x)2614 4163 y Fy([)p Ft(a)2687 4177 y FF(i)2716 4163 y Fy(])20 b Fl(\000)g Ft(a)2900 4177 y FF(i)2928 4163 y Fl(k)26 b Fy(=)f(0)p Fl(g)p Ft(:)448 4367 y Fy(This)c(is)h(naturally)f(a)i Ft(X)7 b Fy(-algebra)23 b(whic)m(h)f(con)m(tains)2208 4299 y Fm(L)2309 4394 y FF(i)p FG(2)p FF(I)2436 4367 y Fn(A)2509 4381 y FF(i)2560 4367 y Fy(as)g(a)h(stable)g(subalgebra,)448 4480 y(so)560 4412 y Fm(L)661 4507 y FF(i)p FG(2)p FF(I)787 4480 y Fn(A)860 4494 y FF(i)919 4480 y Fy(b)s(ecomes)30 b(a)h Ft(X)7 b Fy(-algebra)31 b(to)s(o.)448 4668 y Fz(Prop)s(osition)37 b(3.9)1193 4594 y Fm(\000)1234 4600 y(L)1335 4695 y FF(i)p FG(2)p FF(I)1461 4668 y Fn(A)1534 4682 y FF(i)1563 4594 y Fm(\001)1625 4668 y Fj(o)20 b Ft(X)1823 4643 y Fl(\030)1823 4672 y Fy(=)1919 4600 y Fm(L)2020 4695 y FF(i)p FG(2)p FF(I)2131 4668 y Fy(\()p Fn(A)2239 4682 y FF(i)2288 4668 y Fj(o)g Ft(X)7 b Fy(\))p Fo(.)448 4855 y Fz(Pro)s(of:)48 b Fy(Denote)41 b Fn(A)63 b Fy(=)1347 4787 y Fm(L)1448 4882 y FF(i)p FG(2)p FF(I)1574 4855 y Fn(A)1647 4869 y FF(i)1676 4855 y Fy(.)69 b(Since)39 b(eac)m(h)i Fn(A)2304 4869 y FF(i)2373 4855 y Fy(is)e(an)h(ideal)f(in)f Fn(A)22 b Fy(,)42 b(w)m(e)f(ha)m(v)m(e)448 4968 y(canonical)d(em)m(b)s(eddings) f(of)h Fs(A)1541 4982 y FF(i)1608 4968 y Fl(\021)g Fn(A)1790 4982 y FF(i)1844 4968 y Fj(o)25 b Ft(X)46 b Fy(as)38 b(ideals)f(in)g 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Fy(=)53 b Fl(F)2499 2612 y FG(\003)2585 2645 y Fy(w)m(e)48 b(get)g(\()p Fl(F)3013 2612 y FG(\000)p Fq(1)3108 2645 y Ft(v)s Fy(\)\()p Ft(x)p Fy(\))54 b(=)448 2684 y Fm(R)491 2790 y FF(X)554 2771 y Fx(\003)595 2758 y Fl(h)-15 b(h)p Ft(x;)15 b(k)s Fl(i)-15 b(i)p Ft(v)s Fy(\()p Ft(k)s Fy(\))15 b(d)r Ft(k)51 b Fy(for)d Ft(v)58 b Fl(2)c Ft(L)1616 2725 y Fq(2)1656 2758 y Fy(\()p Ft(X)1773 2725 y FG(\003)1813 2758 y Fy(\).)94 b(W)-8 b(e)49 b(also)f(recall)f(that)i(the)f(dual)f(group)448 2871 y(\()p Ft(X)565 2838 y FG(\003)606 2871 y Fy(\))641 2838 y FG(\003)720 2871 y Fy(of)41 b Ft(X)916 2838 y FG(\003)995 2871 y Fy(is)f(iden)m(ti\014ed)e(with)h Ft(X)7 b Fy(,)43 b(eac)m(h)e Ft(x)g Fl(2)g Ft(X)48 b Fy(b)s(eing)38 b(seen)j(as)f(a)g(c)m(haracter)448 2983 y(of)i Ft(X)645 2950 y FG(\003)727 2983 y Fy(through)f(the)i(form)m(ula)e Ft(x)p Fy(\()p Ft(k)s Fy(\))k(=)f Ft(k)s Fy(\()p Ft(x)p Fy(\),)i(or)c Fl(h)-15 b(h)p Ft(k)s(;)15 b(x)p Fl(i)-15 b(i)47 b Fy(=)d Fl(h)-15 b(h)p Ft(x;)15 b(k)s Fl(i)-15 b(i)p 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b(One)34 b(has)g(a)g(similar)e(iden)m(ti\014cation)h(of)h(the)448 4863 y(group)27 b Ft(C)775 4830 y FG(\003)814 4863 y Fy(-algebra)h(of)g Ft(X)1343 4830 y FG(\003)1411 4863 y Fy(with)e Ft(C)1680 4877 y Fq(0)1719 4863 y Fy(\()p Ft(X)7 b Fy(\).)41 b(By)28 b(using)e(the)i(preceding)f(represen)m (tation)448 4976 y(of)i Ft( )s Fy(\()p Ft(P)13 b Fy(\))29 b(one)g(can)g(pro)m(v)m(e)g(the)f(follo)m(wing)f(result)g(\(the)i(pro)s (of)f(giv)m(en)g(in)f([DaG1)r(])i(for)f Ft(X)1897 5225 y Fy(22)p eop %%Page: 23 23 23 22 bop 448 573 a Fy(a)32 b(real)f(v)m(ector)i(space)f(extends)f (with)f(minor)g(mo)s(di\014cations)f(to)j(the)g(case)g(of)g(ab)s(elian) 448 686 y(groups\).)448 873 y Fz(Prop)s(osition)37 b(3.11)46 b Fo(L)-5 b(et)25 b Ft(')h Fl(2)e Ft(C)1636 840 y Fq(u)1629 901 y(b)1679 873 y Fy(\()p Ft(X)7 b Fy(\))26 b Fo(and)g(let)f Ft( )j Fl(2)d Ft(C)2384 887 y Fq(0)2424 873 y Fy(\()p Ft(X)2541 840 y FG(\003)2581 873 y Fy(\))g Fo(b)-5 b(e)24 b(the)h(F)-7 b(ourier)26 b(tr)-5 b(ans-)448 986 y(form)32 b(of)g(a)f(function)g(fr)-5 b(om)32 b Ft(L)1480 953 y Fq(1)1520 986 y Fy(\()p Ft(X)7 b Fy(\))32 b(\()p Fo(the)g(set)f(of)g (such)g Ft( )k Fo(is)c(a)g(dense)g Fl(\003)p Fo(-sub)-5 b(algebr)g(a)33 b(of)448 1099 y Ft(C)513 1113 y Fq(0)553 1099 y Fy(\()p Ft(X)670 1066 y FG(\003)710 1099 y Fy(\)\))p Fo(.)46 b(Then)34 b(for)h(e)-5 b(ach)34 b(numb)-5 b(er)35 b Ft(")28 b(>)f Fy(0)34 b Fo(ther)-5 b(e)35 b(ar)-5 b(e)35 b(p)-5 b(oints)35 b Ft(x)2721 1113 y Fq(1)2760 1099 y Ft(;)15 b(:)g(:)g(:)i(;)e(x)3014 1113 y FF(n)3089 1099 y Fl(2)27 b Ft(X)41 b Fo(and)448 1212 y(functions)33 b Ft( )902 1226 y Fq(1)942 1212 y Ft(;)15 b(:)g(:)g(:)i(;)e( )1203 1226 y FF(n)1275 1212 y Fl(2)25 b Ft(C)1426 1226 y Fq(0)1466 1212 y Fy(\()p Ft(X)1583 1179 y FG(\003)1623 1212 y Fy(\))33 b Fo(such)g(that)1077 1416 y Fl(k)p Ft( )s Fy(\()p Ft(P)13 b Fy(\))p Ft(')p Fy(\()p Ft(Q)p Fy(\))23 b Fl(\000)1640 1348 y Fm(P)1736 1443 y Fq(1)p FG(\024)p FF(i)p FG(\024)p FF(n)1952 1416 y Ft(')p Fy(\()p Ft(Q)e Fy(+)f Ft(x)2282 1430 y 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b(our)g(purp)s(ose)f(is) g(to)i(sho)m(w)448 3054 y(that)g Fn(A)36 b Fj(o)14 b Ft(X)34 b Fy(can)28 b(also)f(b)s(e)g(realized)f(as)i(a)g Ft(C)1993 3021 y FG(\003)2032 3054 y Fy(-algebra)f(of)h(op)s(erators)f (on)g(the)h(Hilb)s(ert)448 3167 y(space)j Ft(L)753 3134 y Fq(2)793 3167 y Fy(\()p Ft(X)7 b Fy(\).)448 3354 y Fz(Theorem)34 b(3.12)47 b Fo(L)-5 b(et)37 b Fn(A)59 b Fo(b)-5 b(e)38 b(a)g Ft(C)1690 3321 y FG(\003)1728 3354 y Fo(-sub)-5 b(algebr)g(a)39 b(of)f Ft(C)2381 3321 y Fq(u)2374 3382 y(b)2423 3354 y Fy(\()p Ft(X)7 b Fy(\))39 b Fo(stable)f(under)g(tr)-5 b(ansla-)448 3467 y(tions.)42 b(Then)32 b(the)f(line)-5 b(ar)32 b(subsp)-5 b(ac)g(e)32 b Fy([)-12 b([)p Fn(A)38 b Fl(\001)17 b Ft(C)1969 3481 y Fq(0)2008 3467 y Fy(\()p Ft(X)2125 3434 y FG(\003)2166 3467 y Fy(\)])-12 b(])32 b Fo(is)e(a)i Ft(C)2516 3434 y FG(\003)2555 3467 y Fo(-algebr)-5 b(a)31 b(on)h(the)f(Hilb)-5 b(ert)448 3580 y(sp)g(ac)g(e)34 b Ft(L)747 3547 y Fq(2)787 3580 y Fy(\()p Ft(X)7 b Fy(\))33 b Fo(and)448 3784 y Fy(\(3.1\))838 b([)-12 b([)p Fn(A)42 b Fl(\001)21 b Ft(C)1735 3798 y Fq(0)1774 3784 y Fy(\()p Ft(X)1891 3747 y FG(\003)1931 3784 y Fy(\)])-12 b(])2031 3759 y Fl(\030)2031 3789 y Fy(=)2127 3784 y Fn(A)41 b Fj(o)20 b Ft(X)448 3989 y Fo(in)31 b(the)g(sense)f(that)i(ther)-5 b(e)31 b(is)g(a)g(unique)e (isomorphism)34 b Fy(\010)25 b(:)g([)-12 b([)p Fn(A)38 b Fl(\001)15 b Ft(C)2746 4003 y Fq(0)2786 3989 y Fy(\()p Ft(X)2903 3956 y FG(\003)2943 3989 y Fy(\)])-12 b(])26 b Fl(!)g Fn(A)36 b Fj(o)16 b Ft(X)448 4102 y Fo(such)39 b(that)g Fy(\010)15 b([)p Ft(')p Fy(\()p Ft(Q)p Fy(\))p Ft( )s Fy(\()p Ft(P)e Fy(\)])37 b(=)e Ft(S)1586 4117 y FF('; )1742 4102 y Fo(for)k(al)5 b(l)38 b Ft(')e Fl(2)f Fn(A)59 b Fo(and)39 b Ft( )f Fl(2)d Ft(C)2791 4116 y Fq(0)2830 4102 y Fy(\()p Ft(X)2947 4069 y FG(\003)2988 4102 y Fy(\))j Fo(with)3277 4078 y Fm(b)3264 4102 y Ft( )54 b Fl(2)448 4214 y Ft(L)510 4181 y Fq(1)550 4214 y Fy(\()p Ft(X)7 b Fy(\))p Fo(.)54 b(Her)-5 b(e)36 b Ft(S)1061 4229 y FF('; )1215 4214 y Fo(is)g(the)h(element)f Ft(y)f Fl(7!)d Ft(S)2070 4229 y FF('; )2188 4214 y Fy(\()p Fl(\001)p Ft(;)15 b(y)s Fy(\))33 b Fl(2)f Fn(A)57 b Fo(of)37 b Ft(L)2800 4181 y Fq(1)2839 4214 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(A)22 b Fy(\))37 b Fo(de\014ne)-5 b(d)448 4340 y(by)33 b(the)g(function)g Ft(S)1132 4355 y FF('; )1250 4340 y Fy(\()p Ft(x;)15 b(y)s Fy(\))26 b(=)f Ft(')p Fy(\()p Ft(x)p Fy(\))1777 4316 y Fm(b)1763 4340 y Ft( )19 b Fy(\()p Ft(y)s Fy(\))p Fo(.)448 4528 y Fz(Pro)s(of:)48 b Fy(The)24 b(fact)h(that)h([)-12 b([)p Fn(A)30 b Fl(\001)9 b Ft(C)1564 4542 y Fq(0)1603 4528 y Fy(\()p Ft(X)1720 4495 y FG(\003)1761 4528 y Fy(\)])-12 b(])25 b(is)f(a)h Ft(C)2087 4495 y FG(\003)2126 4528 y Fy(-algebra)g(has)f(already)g(b)s(een)g(noticed)448 4641 y(ab)s(o)m(v)m(e,)31 b(but)e(it)g(is)f(also)i(a)f(consequence)h (of)g(the)f(next)h(argumen)m(ts.)41 b(Due)29 b(to)h(Theorem)448 4754 y(3.2)i(w)m(e)f(ha)m(v)m(e)g(the)g(follo)m(wing)e(description)f (of)j Fn(A)41 b Fj(o)20 b Ft(X)7 b Fy(.)41 b(Let)1000 4958 y Fn(H)1089 4972 y FF(X)1182 4958 y Fy(=)25 b Ft(L)1340 4920 y Fq(2)1379 4958 y Fy(\()p Ft(X)7 b Fy(;)15 b Fn(H)28 b Fy(\))1713 4933 y Fl(\030)1713 4962 y Fy(=)1809 4958 y Fn(H)47 b Fl(\012)20 b Ft(L)2098 4920 y Fq(2)2137 4958 y Fy(\()p Ft(X)7 b Fy(\))2316 4933 y Fl(\030)2316 4962 y Fy(=)2412 4958 y Ft(L)2474 4920 y Fq(2)2513 4958 y Fy(\()p Ft(X)28 b Fl(\002)20 b Ft(X)7 b Fy(\))p Ft(:)1897 5225 y Fy(23)p eop %%Page: 24 24 24 23 bop 448 573 a Fy(T)-8 b(o)59 b(eac)m(h)g(in)m(tegrable)f (function)f Ft(S)77 b Fy(:)72 b Ft(X)79 b Fl(!)71 b Fn(A)80 b Fy(w)m(e)58 b(asso)s(ciate)i(an)e(op)s(erator)448 686 y Ft(S)5 b Fl(\017)26 b(2)f Ft(B)5 b Fy(\()p Fn(H)864 700 y FF(X)931 686 y Fy(\))30 b(in)f(the)i(follo)m(wing)e(manner:)40 b(if)29 b Ft(\030)g Fy(:)d Ft(X)32 b Fl(!)26 b Fn(H)56 b Fy(is)30 b Ft(L)2733 653 y Fq(2)2772 686 y Fy(,)h(then)448 912 y(\(3.2\))148 b(\()p Ft(S)26 b Fl(\017)20 b Ft(\030)t Fy(\)\()p Ft(y)s Fy(\))26 b(=)1282 788 y Fm(Z)1333 994 y FF(X)1415 912 y Ft(\034)1455 926 y 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a(24)p eop %%Page: 25 25 25 24 bop 448 573 a Fz(3.6.)155 b Fy(The)42 b Fo(phase)j(sp)-5 b(ac)g(e)44 b Fy(of)f Ft(X)50 b Fy(is)42 b(the)g(lo)s(cally)f(compact)j (ab)s(elian)d(group)h(\004)j(=)448 686 y Ft(X)32 b Fl(\010)25 b Ft(X)733 653 y FG(\003)773 686 y Fy(.)60 b(If)37 b Ft(\030)j Fy(=)c(\()p Ft(x;)15 b(k)s Fy(\))38 b(and)f Ft(\021)i Fy(=)d(\()p Ft(y)s(;)15 b(p)p Fy(\))38 b(are)f(elemen)m(ts)h (of)f(\004,)h(w)m(e)g(set)f Ft(\014)5 b Fy(\()p Ft(\030)t(;)15 b(\021)s Fy(\))38 b(=)448 799 y Fl(h)-15 b(h)p Ft(y)s(;)15 b(k)s Fl(i)-15 b(i)28 b(\001)e(h)-15 b(h\000)p Ft(x;)15 b(p)p Fl(i)-15 b(i)p Fy(.)66 b(Then)38 b Ft(\014)44 b Fy(is)38 b(a)h Fo(bichar)-5 b(acter)41 b Fy(of)d(\004,)j(i.e.)66 b(a)39 b(con)m(tin)m(uous)f(map)h(of)448 912 y(\004)21 b Fl(\002)f Fy(\004)31 b(in)m(to)g Fl(f)p Ft(z)g Fl(2)c Fp(C)50 b Fl(j)27 b(j)p Ft(z)t Fl(j)g Fy(=)g(1)p Fl(g)32 b Fy(whic)m(h)e(is)g(a)i(c)m(haracter)g(in)e(eac)m(h)j(v)-5 b(ariable)30 b(when)g(the)448 1024 y(other)j(one)f(is)f(\014xed.)46 b(Let)32 b Ft(\013)d Fy(:)g(\004)e Fl(!)i Fp(C)56 b Fy(b)s(e)31 b(a)i(con)m(tin)m(uous)e(map)h(suc)m(h)g(that)h Ft(\013)p Fy(\(0\))d(=)e(1)448 1137 y(and)i Fl(j)p Ft(\013)p Fy(\()p Ft(\030)t Fy(\))p Fl(j)d Fy(=)e(1)31 b(for)f(all)f Ft(\030)g Fl(2)c Fy(\004)30 b(and)g(let)g(us)g(set)448 1355 y(\(3.3\))279 b Ft(W)13 b Fy(\()p Ft(\030)t Fy(\))25 b(=)g Ft(\013)p Fy(\()p Ft(\030)t Fy(\))p Ft(U)1480 1369 y FG(\000)p FF(x)1580 1355 y Ft(V)1633 1370 y FF(k)1737 1355 y Fy(and)60 b Ft(!)s Fy(\()p Ft(\030)t(;)15 b(\021)s Fy(\))26 b(=)2472 1294 y Ft(\013)p Fy(\()p Ft(\030)f Fy(+)20 b Ft(\021)s Fy(\))p 2338 1335 601 4 v 2338 1418 a Fl(h)-15 b(h)p Ft(y)s(;)15 b(k)s Fl(i)-15 b(i)p Ft(\013)p Fy(\()p Ft(\030)t Fy(\))p Ft(\013)p Fy(\()p Ft(\021)s Fy(\))2948 1355 y Ft(:)448 1569 y Fy(Then)28 b Ft(W)41 b Fy(is)27 b(a)i(strongly)f(con)m (tin)m(uous)g(map)g(of)g(\004)g(in)m(to)h(the)f(set)h(of)g(unitary)e (op)s(erators)448 1682 y(on)44 b Ft(L)650 1649 y Fq(2)690 1682 y Fy(\()p Ft(X)7 b Fy(\))45 b(satisfying)e Ft(W)13 b Fy(\()p Ft(\030)33 b Fy(+)c Ft(\021)s Fy(\))49 b(=)e Ft(!)s Fy(\()p Ft(\030)t(;)15 b(\021)s Fy(\))p Ft(W)e Fy(\()p Ft(\030)t Fy(\))p Ft(W)g Fy(\()p Ft(\021)s Fy(\).)84 b(In)44 b(other)g(terms,)k Ft(W)448 1795 y Fy(is)d(a)g(pro)5 b(jectiv)m(e)46 b(represen)m(tation)f(of)h(the)f(group)g(\004)g(with)f (phase)h(factor)h Ft(!)i Fy(\(note)448 1908 y(that)40 b Ft(W)13 b Fy(\()p Ft(\030)t Fy(\))p Ft(W)g Fy(\()p Ft(\021)s Fy(\))40 b(=)f Ft(\014)5 b Fy(\()p Ft(\030)t(;)15 b(\021)s Fy(\))p Ft(W)e Fy(\()p Ft(\021)s Fy(\))p Ft(W)g Fy(\()p Ft(\030)t Fy(\)\).)68 b(By)39 b(a)g(w)m(ell)f(kno)m(wn)g (generalization)h(of)448 2021 y(the)34 b(Stone-V)-8 b(on)34 b(Neumann)f(theorem)h(due)e(to)i(G.)g(Mac)m(k)m(ey)i(\(see)e(Theorem)f (C.38)h(in)448 2134 y([RaWi)q(]\),)e(eac)m(h)g(suc)m(h)f(represen)m (tation)h(of)f(\004)g(is)f(unitarily)f(equiv)-5 b(alen)m(t)31 b(to)h(a)f(m)m(ultiple)448 2247 y(of)g Ft(W)43 b Fy(\(whic)m(h)29 b(is)h(irreducible\).)589 2360 y(The)i(represen)m(tation)g Ft(W)44 b Fy(allo)m(ws)31 b(one)i(to)f(construct)h(a)f (pseudo-di\013eren)m(tial)e(cal-)448 2473 y(culus)40 b(whic)m(h)g(dep)s(ends)f(on)i Ft(\013)h Fy(\(di\013eren)m(t)f(c)m (hoices)h(giv)m(e)g(di\013eren)m(t)e(\\quan)m(tization")448 2586 y(rules\).)f(In)29 b(this)e(\014rst)i(part)f(of)h(our)g(pap)s(er)f (the)h(c)m(hoice)g(of)g Ft(\013)h Fy(do)s(es)e(not)h(pla)m(y)g(an)m(y)g (role,)448 2698 y(so)34 b(w)m(e)h(shall)d(not)i(sp)s(ecify)e(it.)51 b(Ho)m(w)m(ev)m(er,)37 b(in)32 b(the)i(second)g(part)g([GI2)q(])g(of)g (the)g(pap)s(er,)448 2811 y(when)g Ft(X)41 b Fy(is)34 b(a)g(v)m(ector)i(space,)g(w)m(e)f(shall)e(\014nd)f(con)m(v)m(enien)m (t)k(to)f(w)m(ork)f(with)f(the)i(W)-8 b(eyl)448 2924 y(calculus)27 b(\(although)i(the)f(algebras)h(w)m(e)f(construct)h(do)g (not)f(dep)s(end)f(on)h(this)g(c)m(hoice\),)448 3037 y(so)j(w)m(e)g(in)m(tro)s(duce)e(here)h(the)h(notations)f(sp)s (eci\014c)g(to)h(this)e(situation.)589 3150 y(Assume)23 b(that)g Ft(X)31 b Fy(is)22 b(a)h(\(real)g(or)g Ft(p)p Fy(-adic\))g(\014nite)f(dimensional)e(v)m(ector)k(space.)39 b(Then)448 3263 y(w)m(e)29 b(equip)f(\004)g(with)f(the)i(symplectic)e (form)h(\(i.e.)h(bilinear)d(an)m(tisymmetric)i(nondegen-)448 3376 y(erate)i(form\))f Ft(\033)j Fy(de\014ned)27 b(b)m(y)i Ft(\033)s Fy(\()p Ft(\030)t(;)15 b(\021)s Fy(\))27 b(=)e Fl(h)p Ft(y)s(;)15 b(k)s Fl(i)j(\000)e(h)p Ft(x;)f(p)p Fl(i)p Fy(.)41 b(Then)28 b Ft(\014)i Fy(=)25 b(e)2864 3343 y FF(i\033)2963 3376 y Fy(and)j(w)m(e)h(tak)m(e)448 3503 y Ft(\013)p Fy(\()p Ft(\030)t Fy(\))e(=)e(e)797 3440 y Fv(i)p 793 3452 31 3 v 793 3493 a Fk(2)833 3467 y FG(h)p FF(x;k)r FG(i)990 3503 y Fy(,)j(whic)m(h)d(giv)m(es)j Ft(!)g Fy(=)d(e)1740 3467 y FG(\000)1809 3440 y Fv(i)p 1805 3452 V 1805 3493 a Fk(2)1845 3467 y FF(\033)1892 3503 y Fy(.)40 b(It)26 b(is)g(con)m(v)m(enien)m(t)i(in)d(this)h(case)i (to)f(use)g(the)448 3616 y(quan)m(tum)j(mec)m(hanical)g(con)m(v)m(en)m (tions)i(and)e(to)h(write)448 3786 y(\(3.4\))757 b Ft(U)1452 3800 y FF(x)1521 3786 y Fy(=)25 b(e)1658 3749 y FF(i)p FG(h)p FF(x;P)10 b FG(i)1855 3786 y Ft(;)136 b(V)2069 3801 y FF(k)2137 3786 y Fy(=)25 b(e)2273 3749 y FF(i)p FG(h)p FF(Q;k)r FG(i)2470 3786 y Ft(:)448 3956 y Fy(Th)m(us)30 b(w)m(e)g(will)e(ha)m(v)m(e)579 4127 y Ft(W)13 b Fy(\()p Ft(\030)t Fy(\))26 b(=)f(e)968 4062 y Fv(i)p 964 4074 V 964 4115 a Fk(2)1005 4089 y FG(h)p FF(x;k)r FG(i)1161 4127 y Fy(e)1202 4089 y FG(\000)p FF(i)p FG(h)p FF(x;P)10 b FG(i)1454 4127 y Fy(e)1494 4089 y FF(i)p FG(h)p FF(Q;k)r FG(i)1716 4127 y Fy(=)25 b(e)1852 4089 y FG(\000)1921 4062 y Fv(i)p 1917 4074 V 1917 4115 a Fk(2)1958 4089 y FG(h)p FF(x;k)r FG(i)2115 4127 y Fy(e)2155 4089 y FF(i)p FG(h)p FF(Q;k)r FG(i)2352 4127 y Fy(e)2392 4089 y FG(\000)p FF(i)p FG(h)p FF(x;P)10 b FG(i)2669 4127 y Fy(=)25 b(e)2805 4089 y FF(i)p Fq(\()p FG(h)p FF(Q;k)r FG(i\000h)p FF(x;P)10 b FG(i)p Fq(\))3281 4127 y Ft(:)448 4297 y Fy(More)31 b(explicitly)-8 b(,)29 b(for)h Ft(f)35 b Fl(2)25 b Ft(L)1462 4264 y Fq(2)1501 4297 y Fy(\()p Ft(X)7 b Fy(\))31 b(w)m(e)g(ha)m(v)m(e) 448 4467 y(\(3.5\))676 b(\()p Ft(W)13 b Fy(\()p Ft(\030)t Fy(\))p Ft(f)d Fy(\)\()p Ft(y)s Fy(\))26 b(=)f(e)1927 4430 y FF(i)p FG(h)p FF(y)r FG(\000)2080 4403 y Fv(x)p 2081 4415 35 3 v 2083 4456 a Fk(2)2126 4430 y FF(;k)r FG(i)2215 4467 y Ft(f)10 b Fy(\()p Ft(y)23 b Fl(\000)d Ft(x)p Fy(\))p Ft(:)589 4638 y Fy(W)-8 b(e)33 b(state)g(no)m(w)e(a)h(v) m(ersion)f(of)h(the)f(Riesz-Kolmogoro)m(v)i(compacit)m(y)f(criterion)e (in)448 4751 y Ft(L)510 4718 y Fq(2)550 4751 y Fy(\()p Ft(X)7 b Fy(\))29 b(\(see)h([GI1)q(]\))f(whic)m(h)f(ma)m(y)h(b)s(e)f (considered)g(as)h(a)g(geometric)h(\(or)f(phase)f(space\))448 4863 y(description)j(of)i Fp(K)15 b Fy(\()q Ft(X)7 b Fy(\).)55 b(W)-8 b(e)34 b(shall)d(adopt)i(the)g(follo)m(wing)e (abbreviation:)45 b(if)31 b Ft(S)38 b Fy(is)32 b(an)448 4976 y(op)s(erator)f(on)f Ft(L)1003 4943 y Fq(2)1043 4976 y Fy(\()p Ft(X)7 b Fy(\))31 b(then)f Ft(S)1494 4943 y FG(?)1578 4976 y Fy(=)25 b(1)c Fl(\000)f Ft(S)5 b Fy(.)1897 5225 y(25)p eop %%Page: 26 26 26 25 bop 448 573 a Fz(Prop)s(osition)37 b(3.13)46 b Fo(A)n(n)34 b(op)-5 b(er)g(ator)37 b Ft(T)42 b Fl(2)28 b Fp(B)14 b Fy(\()p Ft(X)8 b Fy(\))41 b Fo(is)34 b(c)-5 b(omp)g(act)37 b(if)d(and)i(only)f(if)f(it)h(sat-)448 686 y(is\014es)e(one)g(of)g(the)g(fol)5 b(lowing)33 b(e)-5 b(quivalent)33 b(c)-5 b(onditions:)448 799 y Fy(\(i\))33 b(lim)703 813 y FF(x)p FG(!)p Fq(0)868 799 y Fl(k)p Fy(\()p Ft(U)1010 813 y FF(x)1075 799 y Fl(\000)20 b Fy(1\))p Ft(T)13 b Fl(k)26 b Fy(=)f(0)33 b Fo(and)h Fy(lim)1860 814 y FF(k)r FG(!)p Fq(0)2024 799 y Fl(k)p Fy(\()p Ft(V)2157 814 y FF(k)2220 799 y Fl(\000)20 b Fy(1\))p Ft(T)13 b Fl(k)26 b Fy(=)f(0)p Fo(.)448 912 y Fy(\(ii\))32 b(lim)728 927 y FF(\030)s FG(!)p Fq(0)887 912 y Fl(k)p Fy(\()p Ft(W)13 b Fy(\()p Ft(\030)t Fy(\))21 b Fl(\000)f Fy(1\))p Ft(T)13 b Fl(k)26 b Fy(=)f(0)p Fo(.)448 1024 y Fy(\(iii\))f Fl(8)p Ft(")h(>)g Fy(0)h Fl(9)p Ft(')g Fl(2)f Ft(C)1192 1038 y Fq(c)1227 1024 y Fy(\()p Ft(X)7 b Fy(\))27 b Fl(9)p Ft( )h Fl(2)d Ft(C)1695 1038 y Fq(c)1730 1024 y Fy(\()p Ft(X)1847 992 y FG(\003)1888 1024 y Fy(\))g Fo(such)h(that)h Fl(k)p Ft(')p Fy(\()p Ft(Q)p Fy(\))2573 992 y FG(?)2634 1024 y Ft(T)13 b Fl(k)5 b Fy(+)g Fl(k)p Ft( )s Fy(\()p Ft(P)13 b Fy(\))3074 992 y FG(?)3134 1024 y Ft(T)g Fl(k)26 b Ft(<)f(")p Fo(.)448 1212 y Fy(Tw)m(o)31 b(other)f(descriptions)f(of)h (the)h(algebra)f Fp(K)15 b Fy(\()q Ft(X)7 b Fy(\))37 b(are)31 b(furnished)c(b)m(y)j(the)h(result:)448 1400 y Fz(Prop)s(osition)37 b(3.14)46 b Fp(K)15 b Fy(\()p Ft(X)8 b Fy(\))31 b(=)25 b([)-12 b([)p Ft(C)1693 1414 y Fq(0)1733 1400 y Fy(\()p Ft(X)7 b Fy(\))22 b Fl(\001)e Ft(C)2017 1414 y Fq(0)2057 1400 y Fy(\()p Ft(X)2174 1367 y FG(\003)2214 1400 y Fy(\)])-12 b(])2313 1375 y Fl(\030)2313 1404 y Fy(=)2409 1400 y Ft(C)2474 1414 y Fq(0)2513 1400 y Fy(\()p Ft(X)7 b Fy(\))22 b Fj(o)e Ft(X)r(:)448 1587 y Fy(F)-8 b(or)27 b(the)g(pro)s(of)e(of)i(the)f(\014rst)f(equalit)m(y) -8 b(,)28 b(see)f([GI1])g(for)f(example.)39 b(Then)25 b(the)h(canonical)448 1700 y(isomorphism)f(with)h Ft(C)1246 1714 y Fq(0)1286 1700 y Fy(\()p Ft(X)7 b Fy(\))15 b Fj(o)g Ft(X)35 b Fy(follo)m(ws)27 b(from)g(Theorem)g(3.12)i(\(for)f(another)g (pro)s(of)448 1813 y(of)j(the)f(isomorphism)e(of)i Fp(K)15 b Fy(\()q Ft(X)7 b Fy(\))37 b(with)29 b Ft(C)1870 1827 y Fq(0)1909 1813 y Fy(\()p Ft(X)7 b Fy(\))21 b Fj(o)f Ft(X)38 b Fy(see)31 b(Prop)s(osition)d(3.3)j(in)e([T)-8 b(ak)q(]\).)448 2004 y Fz(3.7.)115 b Fy(Later)37 b(on)e(w)m(e)i(will)c (need)j(the)g(follo)m(wing)e(noncomm)m(utativ)m(e)j(v)m(ersion)f(of)g (the)448 2117 y(Ascoli)30 b(theorem)h(\(whic)m(h)e(follo)m(ws)h(from)g (Prop)s(osition)e(3.13,)k(see)f([GI1)q(])g(for)f(details\).)448 2305 y Fz(Prop)s(osition)37 b(3.15)46 b Fo(A)d(b)-5 b(ounde)g(d)45 b(subset)e Fn(K)74 b Fl(\032)45 b Fp(B)13 b Fy(\()q Ft(X)7 b Fy(\))50 b Fo(is)43 b(a)h(r)-5 b(elatively)45 b(c)-5 b(omp)g(act)448 2418 y(set)40 b(of)g(c)-5 b(omp)g(act)42 b(op)-5 b(er)g(ators)43 b(if)c(and)i(only)f(if)g(it)f(satis\014es)i (the)f(fol)5 b(lowing)41 b(e)-5 b(quivalent)448 2531 y(c)g(onditions:)448 2644 y Fy(\(i\))33 b(lim)577 2698 y Fv(x)p Fx(!)p Fk(0)731 2644 y Fy(sup)718 2716 y Fv(T)8 b Fx(2)p Fe(K)881 2644 y Fl(k)p Fy(\()p Ft(U)1023 2658 y FF(x)1088 2644 y Fl(\000)20 b Fy(1\))p Ft(T)1325 2611 y Fq(\()p FG(\003)p Fq(\))1420 2644 y Fl(k)25 b Fy(=)g(0)33 b Fo(and)82 b Fy(lim)1889 2698 y Fv(k)q Fx(!)p Fk(0)2043 2644 y Fy(sup)2030 2716 y Fv(T)8 b Fx(2)p Fe(K)2193 2644 y Fl(k)p Fy(\()p Ft(V)2326 2659 y FF(k)2390 2644 y Fl(\000)20 b Fy(1\))p Ft(T)2627 2611 y Fq(\()p FG(\003)p Fq(\))2721 2644 y Fl(k)26 b Fy(=)f(0)p Ft(:)448 2820 y Fy(\(ii\))32 b(lim)728 2835 y FF(\030)s FG(!)p Fq(0)887 2820 y Fy(sup)1024 2842 y FF(T)10 b FG(2)p Ff(K)1230 2820 y Fl(k)p Fy(\()p Ft(W)j Fy(\()p Ft(\030)t Fy(\))22 b Fl(\000)e Fy(1\))p Ft(T)1782 2787 y Fq(\()p FG(\003)p Fq(\))1876 2820 y Fl(k)26 b Fy(=)f(0)p Ft(:)448 2933 y Fy(\(iii\))14 b Fo(F)-7 b(or)34 b(e)-5 b(ach)33 b Ft(")26 b(>)f Fy(0)33 b Fo(ther)-5 b(e)34 b(ar)-5 b(e)33 b Ft(')26 b Fl(2)f Ft(C)1843 2947 y Fq(c)1878 2933 y Fy(\()p Ft(X)7 b Fy(\))34 b Fo(and)f Ft( )c Fl(2)c Ft(C)2479 2947 y Fq(c)2514 2933 y Fy(\()p Ft(X)2631 2900 y FG(\003)2672 2933 y Fy(\))33 b Fo(such)f(that)946 3138 y Fl(k)p Ft(')p Fy(\()p Ft(Q)p Fy(\))1192 3100 y FG(?)1253 3138 y Ft(T)1319 3100 y Fq(\()p FG(\003)p Fq(\))1413 3138 y Fl(k)21 b Fy(+)e Fl(k)p Ft( )s Fy(\()p Ft(P)13 b Fy(\))1817 3100 y FG(?)1878 3138 y Ft(T)1944 3100 y Fq(\()p FG(\003)p Fq(\))2038 3138 y Fl(k)26 b Ft(<)f(")65 b(f)10 b(or)35 b(al)r(l)f(T)k Fl(2)25 b Fn(K)k Ft(:)589 3342 y Fy(Finally)-8 b(,)41 b(w)m(e)f(pro)m(v)m(e)g (a)g(result)e(men)m(tioned)h(in)f(the)i(In)m(tro)s(duction)e(whic)m(h)g (is)h(the)448 3455 y(main)28 b(justi\014cation)g(for)g(considering)f (crossed)i(pro)s(ducts)f(as)h Ft(C)2653 3422 y FG(\003)2692 3455 y Fy(-algebras)g(of)g(energy)448 3568 y(observ)-5 b(ables.)448 3755 y Fz(Prop)s(osition)37 b(3.16)46 b Fo(L)-5 b(et)33 b Fn(A)54 b Fo(b)-5 b(e)33 b(a)g Ft(C)1792 3722 y FG(\003)1831 3755 y Fo(-sub)-5 b(algebr)g(a)34 b(of)f Ft(C)2474 3722 y Fq(u)2467 3783 y(b)2517 3755 y Fy(\()p Ft(X)7 b Fy(\))33 b Fo(which)h(c)-5 b(ontains)35 b(the)448 3868 y(c)-5 b(onstants)35 b(and)f(is)g(stable)f(under)h(tr)-5 b(anslations.)46 b(L)-5 b(et)34 b Ft(h)26 b Fy(:)g Ft(X)2538 3835 y FG(\003)2604 3868 y Fl(!)h Fp(R)41 b Fo(b)-5 b(e)33 b(a)h(c)-5 b(ontinuous)448 3981 y(non-c)g(onstant)28 b(function)e(such)g(that)h Fy(lim)1842 3996 y FF(k)r FG(!1)2041 3981 y Fl(j)p Ft(h)p Fy(\()p Ft(k)s Fy(\))p Fl(j)g Fy(=)e Fl(1)p Fo(.)39 b(Then)26 b Fn(A)h Fj(o)6 b Ft(X)33 b Fo(is)25 b(the)i Ft(C)3366 3948 y FG(\003)3405 3981 y Fo(-)448 4094 y(algebr)-5 b(a)32 b(gener)-5 b(ate)g(d)31 b(by)f(the)g(self-adjoint)h(op)-5 b(er)g(ators)34 b(of)c(the)g(form)h Ft(h)p Fy(\()p Ft(P)e Fy(+)15 b Ft(k)s Fy(\))g(+)g Ft(V)k Fy(\()p Ft(Q)p Fy(\))p Fo(,)448 4207 y(with)34 b Ft(k)28 b Fl(2)d Ft(X)889 4174 y FG(\003)962 4207 y Fo(and)33 b Ft(V)46 b Fl(2)24 b Fn(A)54 b Fo(r)-5 b(e)g(al.)448 4395 y Fz(Pro)s(of:)48 b Fy(Let)39 b Fs(C)g Fy(b)s(e)f(the)i Ft(C)1419 4362 y FG(\003)1458 4395 y Fy(-algebra)f(generated)h(b)m(y)f (the)g(op)s(erators)g Ft(H)46 b Fy(=)40 b Ft(h)p Fy(\()p Ft(P)f Fy(+)448 4507 y Ft(k)s Fy(\))20 b(+)f Ft(V)h Fy(\()p Ft(Q)p Fy(\))26 b Fl(\021)f Ft(H)1056 4521 y Fq(0)1114 4507 y Fy(+)19 b Ft(V)i Fy(\()p Ft(Q)p Fy(\),)30 b(with)f Ft(k)f Fl(2)d Ft(X)1925 4474 y FG(\003)1995 4507 y Fy(and)k Ft(V)46 b Fl(2)25 b Fn(A)51 b Fy(real.)40 b(By)30 b(making)f(a)i(norm) 448 4620 y(con)m(v)m(ergen)m(t)i(series)d(expansion)f(for)h(large)h Ft(z)1019 4833 y Fy(\()p Ft(z)25 b Fl(\000)20 b Ft(H)7 b Fy(\))1330 4796 y FG(\000)p Fq(1)1449 4833 y Fy(=)1546 4747 y Fm(X)1545 4943 y FF(n)p FG(\025)p Fq(0)1678 4833 y Fy(\()p Ft(z)25 b Fl(\000)20 b Ft(H)1947 4847 y Fq(0)1986 4833 y Fy(\))2021 4796 y FG(\000)p Fq(1)2116 4833 y Fy([)p Ft(V)g Fy(\()p Ft(Q)p Fy(\)\()p Ft(z)26 b Fl(\000)20 b Ft(H)2626 4847 y Fq(0)2665 4833 y Fy(\))2700 4796 y FG(\000)p Fq(1)2794 4833 y Fy(])2819 4796 y FF(n)1897 5225 y Fy(26)p eop %%Page: 27 27 27 26 bop 448 573 a Fy(w)m(e)27 b(get)f Fs(C)g Fl(\032)f Fn(A)32 b Fj(o)11 b Ft(X)c Fy(.)39 b(It)26 b(remains)f(to)h(pro)m(v)m (e)h(the)f(opp)s(osite)f(inclusion.)36 b(F)-8 b(or)27 b(eac)m(h)f Ft(\026)g Fl(2)448 686 y Fp(R)31 b Fy(the)21 b(op)s(erator)h Ft(H)1117 700 y FF(\026)1188 686 y Fy(=)j Ft(h)p Fy(\()p Ft(P)15 b Fy(+)r Ft(k)s Fy(\))r(+)r Ft(\026V)23 b Fy(\()p Ft(Q)p Fy(\))f(is)e(a\016liated)h(to)h Fs(C)g Fy(and)e(the)i(function)e Ft(\026)25 b Fl(7!)448 799 y Fy(\()p Ft(H)559 813 y FF(\026)612 799 y Fl(\000)6 b Ft(i)p Fy(\))755 766 y FG(\000)p Fq(1)874 799 y Fy(is)22 b(norm)h(deriv)-5 b(able)22 b(at)i Ft(\026)h Fy(=)g(0)f(with)e(deriv)-5 b(ativ)m(e)23 b Fl(\000)p Fy(\()p Ft(H)2710 813 y Fq(0)2756 799 y Fl(\000)6 b Ft(i)p Fy(\))2899 766 y FG(\000)p Fq(1)2994 799 y Ft(V)20 b Fy(\()p Ft(Q)p Fy(\)\()p Ft(H)3320 813 y Fq(0)3366 799 y Fl(\000)448 912 y Ft(i)p Fy(\))514 879 y FG(\000)p Fq(1)609 912 y Fy(.)50 b(Hence)35 b(\()p Ft(H)1069 926 y Fq(0)1130 912 y Fl(\000)23 b Ft(i)p Fy(\))1290 879 y FG(\000)p Fq(1)1384 912 y Ft(V)e Fy(\()p Ft(Q)p Fy(\)\()p Ft(H)1711 926 y Fq(0)1773 912 y Fl(\000)h Ft(i)p Fy(\))1932 879 y FG(\000)p Fq(1)2058 912 y Fl(2)30 b Fs(C)p Fy(.)50 b(Let)34 b Ft(\022)f Fl(2)d Ft(C)2683 926 y Fq(c)2718 912 y Fy(\()p Fp(R)s Fy(\))40 b(with)33 b Ft(\022)s Fy(\(0\))d(=)h(1)448 1024 y(and)j Ft(")e(>)g Fy(0.)53 b(Since)33 b Ft(H)1246 1038 y Fq(0)1319 1024 y Fy(is)h(a\016liated)g(to)h Fs(C)p Fy(,)g(w)m(e)g(get)g Ft(\022)s Fy(\()p Ft("H)2521 1038 y Fq(0)2560 1024 y Fy(\)\()p Ft(H)2706 1038 y Fq(0)2769 1024 y Fl(\000)23 b Ft(i)p Fy(\))32 b Fl(2)g Fs(C)p Fy(,)j(and)f(so)448 1137 y Ft(\022)s Fy(\()p Ft("H)647 1151 y Fq(0)686 1137 y Fy(\))p Ft(V)21 b Fy(\()p Ft(Q)p Fy(\)\()p Ft(H)1048 1151 y Fq(0)1111 1137 y Fl(\000)i Ft(i)p Fy(\))1271 1104 y FG(\000)p Fq(1)1400 1137 y Fl(2)33 b Fs(C)p Fy(.)55 b(F)-8 b(rom)35 b(the)h(uniform)d(con)m(tin)m(uit)m(y)i(of)g Ft(V)21 b Fy(,)36 b(and)f(since)448 1250 y(\()p Ft(h)p Fy(\()p Ft(p)p Fy(+)p Ft(k)s Fy(\))p Fl(\000)p Ft(i)p Fy(\))909 1217 y FG(\000)p Fq(1)1031 1250 y Fl(!)25 b Fy(0)20 b(when)g Ft(p)25 b Fl(!)g(1)20 b Fy(in)f Ft(X)1916 1217 y FG(\003)1956 1250 y Fy(,)j(w)m(e)f(get)g Fl(k)p Fy(\()p Ft(U)2410 1264 y FF(x)2455 1250 y Fl(\000)p Fy(1\))p Ft(V)g Fy(\()p Ft(Q)p Fy(\)\()p Ft(H)2933 1264 y Fq(0)2973 1250 y Fl(\000)p Ft(i)p Fy(\))3110 1217 y FG(\000)p Fq(1)3205 1250 y Fl(k)k(!)h Fy(0)448 1363 y(if)d Ft(x)i Fl(!)g Fy(0)f(in)f Ft(X)7 b Fy(.)39 b(This)22 b(implies)e(lim)1662 1377 y FF(")p FG(!)p Fq(0)1819 1363 y Ft(\022)s Fy(\()p Ft("H)2018 1377 y Fq(0)2057 1363 y Fy(\))p Ft(V)h Fy(\()p Ft(Q)p Fy(\)\()p Ft(H)2419 1377 y Fq(0)2466 1363 y Fl(\000)7 b Ft(i)p Fy(\))2610 1330 y FG(\000)p Fq(1)2730 1363 y Fy(=)25 b Ft(V)20 b Fy(\()p Ft(Q)p Fy(\)\()p Ft(H)3152 1377 y Fq(0)3199 1363 y Fl(\000)7 b Ft(i)p Fy(\))3343 1330 y FG(\000)p Fq(1)448 1476 y Fy(in)22 b(norm)h(in)f Ft(B)5 b Fy(\()p Ft(L)1048 1443 y Fq(2)1087 1476 y Fy(\()p Ft(X)i Fy(\)\))25 b(\(indeed,)f(for)f Ft(T)38 b Fl(2)25 b Fp(B)14 b Fy(\()p Ft(X)7 b Fy(\))30 b(w)m(e)24 b(ha)m(v)m(e)h(lim) 2644 1490 y FF(x)p FG(!)p Fq(0)2809 1476 y Fl(k)p Fy(\()p Ft(U)2951 1490 y FF(x)3002 1476 y Fl(\000)6 b Fy(1\))p Ft(T)13 b Fl(k)26 b Fy(=)f(0)448 1589 y(if)31 b(and)f(only)h(if)f(for)h (eac)m(h)i Ft(\016)d(>)d Fy(0)32 b(there)f(is)g Ft(\021)f Fl(2)c Ft(C)2139 1603 y Fq(c)2175 1589 y Fy(\()p Ft(X)2292 1556 y FG(\003)2332 1589 y Fy(\))32 b(suc)m(h)f(that)g Fl(k)p Ft(\021)s Fy(\()p Ft(P)13 b Fy(\))3036 1556 y FG(?)3097 1589 y Ft(T)g Fl(k)27 b Ft(<)g(\016)s Fy(\).)448 1702 y(Hence)37 b Ft(V)20 b Fy(\()p Ft(Q)p Fy(\)\()p Ft(h)p Fy(\()p Ft(P)39 b Fy(+)24 b Ft(k)s Fy(\))g Fl(\000)g Ft(i)p Fy(\))1523 1669 y FG(\000)p Fq(1)1653 1702 y Fl(2)34 b Fs(C)i Fy(for)g(eac)m(h)h Ft(k)h Fl(2)d Ft(X)2463 1669 y FG(\003)2539 1702 y Fy(and)g(eac)m(h)i Ft(V)56 b Fl(2)34 b Fn(A)57 b Fy(real.)448 1815 y(But)33 b Ft(H)707 1829 y Fq(0)778 1815 y Fy(is)e(a\016liated)h(to)h Fs(C)p Fy(,)g(so)f(this)f (implies)f Ft(')p Fy(\()p Ft(Q)p Fy(\))p Ft( )s Fy(\()p Ft(P)13 b Fy(\))30 b Fl(2)e Fs(C)k Fy(for)g(all)f Ft(')e Fl(2)f Fn(A)53 b Fy(\(not)448 1928 y(necessarily)36 b(real\))h(and)g (all)e Ft( )41 b Fy(in)36 b(the)h Fl(\003)p Fy(-subalgebra)f Fn(B)k Fl(\032)c Ft(C)2651 1942 y Fq(0)2691 1928 y Fy(\()p Ft(X)2808 1895 y FG(\003)2848 1928 y Fy(\))h(generated)h(b)m(y)448 2041 y(functions)31 b(of)h(the)g(form)f Ft(p)d Fl(7!)g Ft(\030)t Fy(\()p Ft(h)p Fy(\()p Ft(p)22 b Fy(+)f Ft(k)s Fy(\)\))33 b(with)d Ft(\030)i Fl(2)27 b Ft(C)2426 2055 y FF(c)2461 2041 y Fy(\()p Fp(R)s Fy(\))39 b(and)31 b Ft(k)g Fl(2)c Ft(X)3056 2008 y FG(\003)3096 2041 y Fy(.)46 b(By)32 b(the)448 2154 y(Stone-W)-8 b(eierstrass)36 b(theorem,)h Fn(B)h Fy(is)c(dense)h(in)e Ft(C)2217 2168 y Fq(0)2256 2154 y Fy(\()p Ft(X)2373 2121 y FG(\003)2414 2154 y Fy(\).)54 b(Hence,)37 b(since)d(the)h(set)h(of)448 2267 y Ft( )j Fl(2)c Ft(C)707 2281 y Fq(0)746 2267 y Fy(\()p Ft(X)863 2234 y FG(\003)903 2267 y Fy(\))i(suc)m(h)f(that)h Ft(')p Fy(\()p Ft(Q)p Fy(\))p Ft( )s Fy(\()p Ft(P)13 b Fy(\))37 b Fl(2)e Fs(C)i Fy(is)e(norm)h(closed)g(and)f(con)m(tains)i Fn(B)t Fy(,)h(w)m(e)448 2379 y(\014nally)29 b(obtain)g Ft(')p Fy(\()p Ft(Q)p Fy(\))p Ft( )s Fy(\()p Ft(P)13 b Fy(\))28 b Fl(2)d Fs(C)30 b Fy(for)g(all)f Ft(')d Fl(2)f Fn(A)c Ft(;)15 b( )29 b Fl(2)c Ft(C)2424 2393 y Fq(0)2464 2379 y Fy(\()p Ft(X)2581 2346 y FG(\003)2621 2379 y Fy(\).)p 3371 2371 67 67 v 589 2542 a(The)33 b(prop)s(osition)e(can)i(b)s(e)g (restated)h(as)f(follo)m(ws:)46 b Fn(A)d Fj(o)22 b Ft(X)41 b Fo(is)35 b(the)g(smal)5 b(lest)37 b Ft(C)3366 2509 y FG(\003)3405 2542 y Fo(-)448 2655 y(algebr)-5 b(a)46 b(of)f(op)-5 b(er)g(ators)48 b(on)d Ft(L)1498 2622 y Fq(2)1537 2655 y Fy(\()p Ft(X)7 b Fy(\))46 b Fo(which)f(c)-5 b(ontains)46 b(al)5 b(l)45 b(the)g(r)-5 b(esolvents)46 b Fy(\()p Ft(h)p Fy(\()p Ft(P)13 b Fy(\))30 b(+)448 2768 y Ft(V)21 b Fy(\()p Ft(Q)p Fy(\))e(+)g Ft(i)p Fy(\))839 2735 y FG(\000)p Fq(1)966 2768 y Fo(with)33 b Ft(V)46 b Fl(2)25 b Fn(A)53 b Fo(r)-5 b(e)g(al)33 b(and)h(which)f(is)f(stable)g (under)h(the)g(automorphisms)448 2881 y Ft(S)e Fl(7!)25 b Ft(V)704 2896 y FF(k)746 2881 y Ft(S)5 b(V)881 2848 y FG(\003)860 2909 y FF(k)950 2881 y Fy(\(see)32 b(\(3.4\)\).)448 3167 y FA(4)135 b(Algebras)46 b(of)f(hamiltonians:)62 b(examples)448 3449 y Fz(4.1.)41 b Fy(In)28 b(this)f(section)i(w)m(e)g (shall)e(describ)s(e)g(sev)m(eral)h(algebras)h(of)g(hamiltonian)d(op)s (era-)448 3561 y(tors)d(of)f(ph)m(ysical)f(in)m(terest.)39 b(Most)23 b(of)f(them)h(are)f(treated)i(in)d(a)i(rather)f(succin)m(t)g (manner)448 3674 y(either)36 b(b)s(ecause)f(they)h(presen)m(t)g(no)g (di\016culties,)f(or)h(b)s(ecause)f(they)h(will)e(b)s(e)h(consid-)448 3787 y(ered)e(in)f(detail)g(in)g(other)h(publications.)46 b(Tw)m(o)33 b(examples)g(are)g(studied)e(in)h(depth)g(in)448 3900 y(Section)40 b(5)g(and)f(in)g(the)h(second)f(part)h([GI2)q(])g(of) g(this)e(pap)s(er.)68 b(Roughly)39 b(sp)s(eaking,)448 4013 y(there)31 b(will)d(b)s(e)h(four)h(t)m(yp)s(es)g(of)h(suc)m(h)f (algebras:)448 4176 y Fz(\(i\))21 b Fy(Crossed)g(pro)s(ducts)f(of)h (translation)g(in)m(v)-5 b(arian)m(t)20 b(unital)g Ft(C)2523 4143 y FG(\003)2562 4176 y Fy(-subalgebras)h(of)g Ft(C)3241 4143 y Fq(u)3234 4203 y(b)3284 4176 y Fy(\()p Ft(X)7 b Fy(\))448 4289 y(b)m(y)31 b(the)g(natural)e(action)i(of)g Ft(X)7 b Fy(.)42 b(These)30 b(corresp)s(ond)g(to)h Ft(Q)p Fy(-anisotropic)f(systems)g(\()p Ft(X)448 4402 y Fy(b)s(eing)f(the)i (con\014guration)f(space\).)448 4564 y Fz(\(ii\))23 b Fy(The)g(simplest)e(t)m(yp)s(e)j(of)g(phase)f(space)h(\(or)f Ft(Q)p Fy(-)p Ft(P)13 b Fy(\))24 b(anisotrop)m(y)-8 b(,)25 b(where)e(the)h(algebra)448 4677 y(is)30 b(\(essen)m(tially\))g(of)g (the)g(form)g([)-12 b([)p Fn(A)42 b Fl(\001)20 b Fn(B)t Fy(])-12 b(],)31 b(where)f Fn(A)51 b Fy(and)30 b Fn(B)k Fy(are)d Ft(C)2804 4644 y FG(\003)2843 4677 y Fy(-subalgebras)e(of)448 4790 y Ft(C)520 4757 y Fq(u)513 4818 y(b)563 4790 y Fy(\()p Ft(X)7 b Fy(\))32 b(and)d Ft(C)995 4757 y Fq(u)988 4818 y(b)1038 4790 y Fy(\()p Ft(X)1155 4757 y FG(\003)1195 4790 y Fy(\))i(resp)s(ectiv)m(ely)-8 b(.)448 4953 y Fz(\(iii\))39 b Fy(F)-8 b(ull)38 b(phase)h(space)g(anisotrop)m(y:)59 b(here)39 b(w)m(e)g(consider)f Ft(C)2643 4920 y FG(\003)2682 4953 y Fy(-algebras)i(naturally)1897 5225 y(27)p eop %%Page: 28 28 28 27 bop 448 573 a Fy(asso)s(ciated)32 b(to)h(the)e(Heisen)m(b)s(erg)g (group,)h(whic)m(h)e(is)h(the)g(simplest)f(t)m(yp)s(e)i(of)g(nilp)s (oten)m(t)448 686 y(group.)38 b(The)23 b(algebra)h(w)m(e)g(construct)g (will)d(b)s(e)h(called)h Fo(gr)-5 b(ade)g(d)28 b(symple)-5 b(ctic)28 b(algebr)-5 b(a)25 b Fy(and)448 799 y(will)h(b)s(e)i(studied) f(in)h(detail)f(in)h(the)g(second)h(part)g(of)f(this)g(pap)s(er.)39 b(This)27 b(also)h(suggests)448 912 y(the)33 b(consideration)f(of)h (algebras)g(asso)s(ciated)g(to)h(unitary)d(represen)m(tations)i(of)g (more)448 1024 y(general)f(groups,)f(whic)m(h)f(w)m(e)i(shall,)e(ho)m (w)m(ev)m(er,)j(not)f(do)f(here.)43 b(More)33 b(sp)s(eci\014cally)-8 b(,)30 b(w)m(e)448 1137 y(ha)m(v)m(e)i(in)d(mind)f(groups)i(related)g (to)h(non-euclidean)e(\(h)m(yp)s(erb)s(olic\))g(geometries.)448 1300 y Fz(\(iv\))j Fy(Algebras)g(of)h(creation-anihilation)e(op)s (erators)h(on)h(a)g(F)-8 b(o)s(c)m(k)34 b(space.)47 b(These)33 b(are)448 1413 y(generated)28 b(b)m(y)f(hamiltonians)d(of)j(quan)m(tum) f(\014elds)g(with)f(a)i(particle)f(n)m(um)m(b)s(er)g(cut-o\013.)448 1526 y(Their)31 b(quotien)m(t)h(with)f(resp)s(ect)h(to)h(the)f(ideal)f (of)i(compact)g(op)s(erators)f(has)g(an)g(esp)s(e-)448 1639 y(cially)25 b(nice)h(form:)38 b(w)m(e)26 b(shall)f(describ)s(e)f (the)j(result)e(here)h(but)f(the)i(pro)s(of)e(will)f(b)s(e)h(giv)m(en) 448 1752 y(elsewhere)30 b(\([GM)q(];)h(see)g(also)f([G)q(]\).)589 1914 y(The)25 b(main)f(emphasis)f(of)i(this)f(pap)s(er)g(is)g(on)g(the) i(tec)m(hnique)e(of)h(crossed)g(pro)s(ducts,)448 2027 y(whic)m(h)g(allo)m(ws)h(one)g(to)h(do)f(the)g(computations)g(at)h(an)f (ab)s(elian)e(lev)m(el.)39 b(Assuming)24 b(that)448 2140 y Ft(X)41 b Fo(is)35 b(non)g(c)-5 b(omp)g(act)p Fy(,)37 b(the)c(strategy)h(is)e(as)i(follo)m(ws.)48 b(W)-8 b(e)34 b(\014rst)e(c)m(ho)s(ose)i(a)g Ft(C)3080 2107 y FG(\003)3119 2140 y Fy(-algebra)448 2253 y Fn(A)22 b Fy(,)34 b(in)m(terpreted)g(as)g (the)g(algebra)g(of)g(classical)f(in)m(teractions)g(\(or)i(p)s(oten)m (tials\),)f(suc)m(h)448 2366 y(that:)448 2570 y(\(4.1\))222 b Ft(C)920 2584 y FG(1)994 2570 y Fy(\()p Ft(X)7 b Fy(\))27 b Fl(\032)e Fn(A)46 b Fl(\032)25 b Ft(C)1556 2533 y Fq(u)1549 2593 y(b)1599 2570 y Fy(\()p Ft(X)7 b Fy(\))31 b(and)f Fn(A)82 b Fy(is)29 b(stable)i(under)d(translations)o Ft(:)448 2775 y Fy(W)-8 b(e)41 b(recall)d(that)i Ft(C)1136 2789 y FG(1)1211 2775 y Fy(\()p Ft(X)7 b Fy(\))41 b(=)f Fp(C)50 b Fy(+)26 b Ft(C)1769 2789 y Fq(0)1809 2775 y Fy(\()p Ft(X)7 b Fy(\),)43 b(so)c Fn(A)61 b Fy(is)38 b(unital)g(\(i.e.)h(it)g(con)m(tains)h(the)448 2888 y(constan)m(t)45 b(functions\))e(and)g Ft(C)1524 2902 y Fq(0)1563 2888 y Fy(\()p Ft(X)7 b Fy(\))45 b(is)d(an)i(ideal)e(in)g(it.)80 b(No)m(w,)48 b(if)43 b(w)m(e)h(can)g(giv)m(e)g(a)448 3000 y(con)m(v)m(enien)m(t)h(description)c(of)i Fn(A)21 b Ft(=C)1709 3014 y Fq(0)1749 3000 y Fy(\()p Ft(X)7 b Fy(\),)48 b(then)43 b(b)m(y)g(using)f(Prop)s(osition)f(3.14)k(and)448 3113 y(Theorem)30 b(3.4)i(w)m(e)f(get)448 3318 y(\(4.2\))577 b([)p Fn(A)41 b Fj(o)20 b Ft(X)7 b Fy(])16 b Ft(=)p Fp(K)f Fy(\()q Ft(X)7 b Fy(\))1888 3293 y Fl(\030)1888 3322 y Fy(=)2014 3318 y([)q Fn(A)21 b Ft(=C)2244 3332 y Fq(0)2284 3318 y Fy(\()p Ft(X)7 b Fy(\)])21 b Fj(o)f Ft(X)r(:)589 3522 y Fy(The)31 b(simplest)f(situation)g(is)h(that)h(when)e Fn(A)49 b Fy(=)26 b Ft(C)2326 3536 y FG(1)2401 3522 y Fy(\()p Ft(X)7 b Fy(\).)45 b(Then)30 b(w)m(e)i(get)h(algebra)448 3635 y Fp(T)p Fy(\()p Ft(X)7 b Fy(\))39 b(of)c(energy)h(observ)-5 b(ables)34 b(corresp)s(onding)f(to)j(a)g(t)m(w)m(o-b)s(o)s(dy)f(quan)m (tum)g(system)448 3748 y(\(or)c(a)g(particle)e(in)g(an)i(external)f (\014eld\):)448 3952 y(\(4.3\))193 b Fp(T)p Fy(\()p Ft(X)7 b Fy(\))28 b(=)d Ft(C)1226 3966 y FG(1)1301 3952 y 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b(able)27 b(a\016liated)f(to)h Ft(C)1437 1521 y Fq(u)1430 1582 y(b)1480 1554 y Fy(\()p Ft(X)7 b Fy(\))13 b Fj(o)g Ft(X)36 b Fy(in)26 b(terms)g(of)h(its)g(\\lo)s (calizations)f(at)i(in\014nit)m(y".)448 1667 y(Detailed)j(pro)s(ofs)e (of)i(the)f(results)f(presen)m(ted)i(b)s(elo)m(w)e(will)f(b)s(e)i (found)f(in)g([GI3)q(].)448 1836 y Fz(Theorem)34 b(4.1)46 b Fo(L)-5 b(et)32 b Ft(H)39 b Fo(b)-5 b(e)31 b(an)h(observable)g (a\016liate)-5 b(d)34 b(to)e(the)g(algebr)-5 b(a)33 b Ft(C)3020 1803 y Fq(u)3013 1864 y(b)3063 1836 y Fy(\()p Ft(X)7 b Fy(\))21 b Fj(o)f Ft(X)7 b Fo(.)448 1949 y(Then)33 b(the)g(str)-5 b(ong)34 b(limit)f Fy(s)15 b(-lim)1533 1963 y FF(x;)p Fi({)1661 1949 y Ft(U)1723 1963 y FF(x)1767 1949 y Ft(H)7 b(U)1922 1916 y FG(\003)1912 1972 y FF(x)1987 1949 y Fy(=)25 b Ft(H)2159 1963 y Fi({)2245 1949 y Fo(exists)33 b(for)g(e)-5 b(ach)33 b(ultr)-5 b(a\014lter)35 b Fj({)h Fo(on)448 2062 y Ft(X)k Fo(\014ner)33 b(than)h(the)f(F)-7 b(r)n(\023)-44 b(echet)33 b(\014lter)h(and)f(one)g(has)448 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Ft(K)48 b Fy(with)40 b Ft(K)50 b Fl(2)44 b Fp(K)15 b Fy(\()p Ft(X)8 b Fy(\))50 b(=)43 b Ft(C)1901 2891 y Fq(0)1940 2877 y Fy(\()p Ft(X)7 b Fy(\))29 b Fj(o)e Ft(X)49 b Fy(and)41 b Ft(')j Fl(2)g Ft(C)2804 2891 y Fq(0)2843 2877 y Fy(\()p Fp(R)s Fy(\))k(\(it)41 b(su\016ces)448 2990 y(in)36 b(fact)h(to)g(ha)m(v)m(e)h(this)e(for)g Ft(')p Fy(\()p Ft(t)p Fy(\))g(=)g(\()p Ft(t)24 b Fl(\006)g Ft(i)p Fy(\))1966 2957 y FG(\000)p Fq(1)2061 2990 y Fy(\).)60 b(W)-8 b(e)38 b(stress)e(that)h(ev)m(en)g(if)f(w)m(e)h(w)m(ork)448 3103 y(in)g(the)h(standard)f(represen)m(tation)h(on)g Ft(L)1911 3070 y Fq(2)1950 3103 y Fy(\()p Ft(X)7 b Fy(\))39 b(and)e Ft(H)45 b Fy(is)37 b(a)h(densely)f(de\014ned)g(self-)448 3216 y(adjoin)m(t)31 b(op)s(erator,)g(the)g(limits)e Ft(H)1636 3230 y Fi({)1720 3216 y Fy(are)i(observ)-5 b(ables)30 b(but)g(are)i(not)f(densely)e(de\014ned)448 3329 y(self-adjoin)m(t)40 b(op)s(erators)h(on)f Ft(L)1537 3296 y Fq(2)1576 3329 y Fy(\()p Ft(X)7 b Fy(\))42 b(in)d(general.)71 b(F)-8 b(or)41 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b(\014lter)h(the)g(limit)f Ft(H)3287 4877 y Fi({)3366 4863 y Fy(=)448 4976 y(s)15 b(-)q(lim)656 4990 y FF(x;)p Fi({)785 4976 y Ft(U)847 4990 y FF(x)891 4976 y Ft(H)7 b(U)1046 4943 y FG(\003)1036 4999 y FF(x)1117 4976 y Fo(exists)33 b(in)g(str)-5 b(ong)34 b(r)-5 b(esolvent)34 b(sense,)e(and)i Fr(\033)2631 4990 y Fq(ess)2722 4976 y Fy(\()p Ft(H)7 b Fy(\))26 b(=)2997 4908 y Fm(S)3073 5003 y Fi({)3142 4976 y Fr(\033)l Fy(\()p Ft(H)3311 4990 y Fi({)3365 4976 y Fy(\))p Ft(:)1897 5225 y Fy(29)p eop %%Page: 30 30 30 29 bop 589 573 a Fy(The)40 b(op)s(erators)g Ft(H)1274 587 y Fi({)1368 573 y Fy(will)d(b)s(e)j(called)f Fo(lo)-5 b(c)g(alizations)45 b(at)d(in\014nity)e Fy(of)g Ft(H)7 b Fy(.)70 b(T)-8 b(o)41 b(b)s(e)448 686 y(precise)28 b(w)m(e)h(should)d(sa)m(y)j Ft(Q)15 b Fy(-in\014nit)m(y)-8 b(.)39 b(Indeed,)28 b(the)h(\\in\014nit)m(y")e(of)h(other)h(observ)-5 b(ables)448 799 y(\(e.g.)26 b(the)e(momen)m(tum)g Ft(P)13 b Fy(\))24 b(could)e(pla)m(y)i(a)g(role)g(to)s(o)g(and)f(in)g(fact)i 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y(b)s(e)i(used)g(to)h(c)m(hec)m(k)h(the)e(a\016liation)f(to)j Ft(C)1978 1669 y Fq(u)1971 1730 y(b)2020 1702 y Fy(\()p Ft(X)7 b Fy(\))31 b Fj(o)e Ft(X)52 b Fy(of)45 b(op)s(erators)f(of)h (the)g(form)448 1815 y Ft(H)33 b Fy(=)24 b Ft(h)p Fy(\()p Ft(P)13 b Fy(\))22 b(+)e Ft(V)g Fy(\()p Ft(Q)p Fy(\).)589 1928 y(Some)29 b(particular)d(situations)i(co)m(v)m(ered)i(b)m(y)e(the) g(preceding)g(corollary)f(ha)m(v)m(e)j(b)s(een)448 2041 y(treated)g(b)m(y)f(di\013eren)m(t)g(metho)s(ds)f(in)g([HeM)q(].)41 b(W)-8 b(e)30 b(men)m(tion)f(only)f(t)m(w)m(o)j(examples,)e(the)448 2154 y(\014rst)42 b(one)g(b)s(eing)f(b)s(orro)m(w)m(ed)h(from)f([HeM)q (].)76 b(Let)43 b Ft(H)52 b Fy(=)44 b(\001)28 b(+)g Ft(V)20 b Fy(\()p Ft(x)p Fy(\))43 b(in)e Ft(L)3100 2121 y Fq(2)3139 2154 y Fy(\()p Fp(R)s Fy(\))q(.)82 b(If)448 2267 y Ft(V)21 b Fy(\()p Ft(x)p Fy(\))k(=)g Fl(h)p Ft(x)p Fl(i)15 b Fy(cos)1025 2234 y Fq(2)1080 2189 y Fm(p)p 1171 2189 123 4 v 78 x Fl(h)p Ft(x)p Fl(i)p Fy(,)31 b(then)f Fr(\033)1610 2281 y Fq(ess)1701 2267 y Fy(\()p Ft(H)7 b Fy(\))26 b(=)f Fp(N)33 b Fy(+)20 b(1)p Ft(=)p Fy(2)32 b(\(here)e Fl(h)p Ft(x)p Fl(i)c Fy(=)f(\(1)c(+)f Fl(j)p Ft(x)p Fl(j)3090 2234 y Fq(2)3130 2267 y Fy(\))3165 2234 y Fq(1)p FF(=)p Fq(2)3275 2267 y Fy(\).)42 b(If)448 2379 y Ft(V)21 b Fy(\()p Ft(x)p Fy(\))k(=)g(sin)892 2302 y Fm(p)p 983 2302 103 4 v 77 x Fl(j)p Ft(x)p Fl(j)q Fy(,)e(then)e Fr(\033)1386 2393 y Fq(ess)1477 2379 y Fy(\()p Ft(H)7 b Fy(\))26 b(=)f([)p Fl(\000)p Fy(1)p Ft(;)15 b Fl(1)p Fy(\).)39 b(T)-8 b(o)22 b(get)g(this)f(it)g(su\016ces)g(to)h(determine) 448 2492 y(the)31 b(op)s(erators)g Ft(H)1084 2506 y Fi({)1137 2492 y Fy(.)41 b(It)31 b(is)f(easy)h(to)g(sho)m(w)f(\(the)h (computations)g(are)g(clari\014ed)d(b)m(y)j(the)448 2605 y(notion)f(of)g(lo)s(calization)f(at)h(in\014nit)m(y)e(for)i(the)g (case)h(of)f(functions)e(in)m(tro)s(duced)g(b)s(elo)m(w\))448 2718 y(that)36 b(in)d(the)i(\014rst)f(case)i(these)g(are)f(exactly)h (the)f(op)s(erators)g(\001)23 b(+)g(\()p Ft(\025)g Fl(\006)g Ft(Q=)p Fy(2\))3185 2685 y Fq(2)3260 2718 y Fy(with)448 2831 y Ft(\025)j Fl(2)e Fp(R)s Fy(,)37 b(while)28 b(for)i(the)h(second) f(example)h(they)f(are)h(\001)20 b(+)g Ft(\025)30 b Fy(with)f Ft(\025)c Fl(2)g Fy([)p Fl(\000)p Fy(1)p Ft(;)15 b Fy(1].)589 2944 y(W)-8 b(e)27 b(shall)d(no)m(w)i(giv)m(e)g(the)g(main)f(ideea)g (of)h(the)g(pro)s(of)f(of)h(Theorem)f(4.1)i(in)d(order)h(to)448 3057 y(mak)m(e)30 b(the)e(connection)h(with)e(the)i(algebraic)f(framew) m(ork)g(of)h(the)f(rest)h(of)g(this)e(pap)s(er.)589 3170 y(An)41 b(op)s(erator)h Ft(S)48 b Fl(2)43 b Fp(B)14 b Fy(\()p Ft(X)8 b Fy(\))47 b(will)39 b(b)s(e)h(called)g Fo(r)-5 b(e)g(gular)43 b Fy(if)d Fl(f)p Ft(U)2687 3184 y FF(x)2732 3170 y Ft(S)2793 3137 y Fq(\()p FG(\003)p Fq(\))2887 3170 y Ft(U)2959 3137 y FG(\003)2949 3192 y FF(x)3042 3170 y Fl(j)j Ft(x)g Fl(2)g Ft(X)7 b Fl(g)448 3283 y Fy(is)39 b(a)h(strongly)g(relativ)m(ely)f(compact)i(part)f(of)g Fp(B)14 b Fy(\()p Ft(X)7 b Fy(\))q(.)75 b(Clearly)38 b(the)i(set)h(of)f(regular)448 3396 y(op)s(erators)25 b(is)f(a)h(unital)e Ft(C)1330 3363 y FG(\003)1368 3396 y Fy(-subalgebra)h(of)h Fp(B)14 b Fy(\()p Ft(X)8 b Fy(\))31 b(whic)m(h)23 b(con)m(tains)i Fp(K)15 b Fy(\()p Ft(X)8 b Fy(\).)45 b(F)-8 b(rom)25 b(the)448 3509 y(Riesz-Kolmogoro)m(v)32 b(compacit)m(y)g(criterion)e(it)g(follo)m(ws)g(easily)g(that)i(it)e (also)h(con)m(tains)448 3621 y Ft(C)520 3588 y Fq(u)513 3649 y(b)563 3621 y Fy(\()p Ft(X)7 b Fy(\))15 b Fj(o)g Ft(X)36 b Fy(\(see)28 b(Prop)s(osition)e(3.15\).)42 b(A)28 b(self-adjoin)m(t)f(op)s(erator)g Ft(H)35 b Fy(on)28 b Ft(L)3076 3588 y Fq(2)3115 3621 y Fy(\()p Ft(X)7 b Fy(\))29 b(will)448 3734 y(b)s(e)h(called)g(regular)f(if)h(it)f(is)h (a\016liated)g(to)h(the)f(algebra)h(of)f(regular)g(op)s(erators.)589 3847 y(W)-8 b(e)37 b(recall)d(no)m(w)h(a)g(result)f(of)h(Stone)g (concerning)g(the)g(Stone-)2778 3824 y(\024)2768 3847 y(Cec)m(h)h(compacti\014-)448 3960 y(cation)i(of)f Ft(X)7 b Fy(:)54 b(if)36 b(\010)h(is)f(a)h(con)m(tin)m(uous)g(map)g(from)f Ft(X)45 b Fy(to)37 b(a)h(Hausdor\013)e(top)s(ological)448 4073 y(space)42 b Ft(Y)61 b Fy(and)40 b(if)g(the)i(range)f(of)g(\010)g (is)f(a)h(relativ)m(ely)g(compact)h(set,)i(then)d(\010)g(has)g(a)448 4186 y(unique)26 b(con)m(tin)m(uous)h(extension)h(to)g Ft(\014)5 b(X)35 b Fy(\(this)27 b(prop)s(ert)m(y)g(is)g(kno)m(wn)g(as)h (the)g(univ)m(ersal)448 4299 y(prop)s(ert)m(y)34 b(of)h Ft(\014)5 b(X)i Fy(\).)54 b(This)33 b(is)g(related)i(to)g(the)g(fact)g (that)g(the)g(limit)d(of)j(\010)f(along)h(an)m(y)448 4412 y(ultra\014lter)26 b(exists)i(\(under)f(the)h(same)g (conditions\).)39 b(Recall)27 b(also)h(that)h Ft(X)35 b Fy(is)27 b(an)h(op)s(en)448 4534 y(dense)e(subset)f(of)h Ft(\014)5 b(X)33 b Fy(\(b)s(ecause)26 b Ft(X)34 b Fy(is)24 b(lo)s(cally)h(compact\))i(and)e(denote)2933 4511 y Fm(f)2930 4534 y Ft(X)48 b Fy(=)24 b Ft(\014)5 b(X)19 b Fl(n)11 b Ft(X)448 4647 y Fy(the)31 b(b)s(oundary)d(of)j Ft(X)38 b Fy(in)29 b Ft(\014)5 b(X)i Fy(.)589 4760 y(If)34 b Ft(S)40 b Fy(is)33 b(a)i(regular)f(op)s(erator)h(then)f(the)g(map)g Ft(x)e Fl(7!)g Ft(U)2485 4774 y FF(x)2529 4760 y Ft(S)5 b(U)2662 4727 y FG(\003)2652 4782 y FF(x)2736 4760 y Fy(is)34 b(con)m(tin)m(uous)g(and)448 4873 y(has)f(relativ)m(ely)g (compact)i(range)e(if)f Fp(B)14 b Fy(\()q Ft(X)7 b Fy(\))40 b(is)32 b(equipp)s(ed)f(with)h(the)h(strong)h(op)s(erator)1897 5225 y(30)p eop %%Page: 31 31 31 30 bop 448 573 a Fy(top)s(ology)-8 b(,)26 b(hence)d(it)g(extends)h (to)g(a)f(strongly)g(con)m(tin)m(uous)g(map)g Fj({)28 b Fl(7!)d Ft(S)2909 587 y Fi({)2987 573 y Fy(from)d Ft(\014)5 b(X)31 b Fy(to)448 686 y Fp(B)14 b Fy(\()q Ft(X)7 b Fy(\).)50 b(The)31 b(op)s(erators)g Ft(S)1378 700 y Fi({)1463 686 y Fy(with)f Fj({)g Fl(2)1853 663 y Fm(f)1849 686 y Ft(X)54 b Fy(will)29 b(b)s(e)i(called)f Fo(lo)-5 b(c)g(alizations)37 b(at)d(in\014nity)448 799 y Fy(of)f Ft(S)5 b Fy(.)48 b(It)33 b(is)f(clear)h(no)m(w)g(ho)m(w)g(to)h(de\014ne)e(the)h(lo)s (calizations)e(at)j(in\014nit)m(y)d(of)i(a)g(regular)448 912 y(\(p)s(ossibly)19 b(un)m(b)s(ounded\))g(self-adjoin)m(t)i(op)s (erator)g(\(of)h(course,)i(these)e(lo)s(calizations)e(will)448 1024 y(not)27 b(b)s(e)e(densely)h(de\014ned)f(op)s(erators)h(in)f (general,)i(so)g(it)f(is)f(more)i(con)m(v)m(enien)m(t)g(to)g(w)m(ork) 448 1137 y(from)k(the)f(b)s(egining)f(with)g(observ)-5 b(ables,)30 b(as)h(w)m(e)g(ha)m(v)m(e)h(done)f(in)e(Theorem)h(4.1\).)44 b(The)448 1250 y(inclusion)33 b Fr(\033)887 1264 y Fq(ess)979 1250 y Fy(\()p Ft(H)7 b Fy(\))35 b Fl(\033)1273 1182 y Fm(S)1348 1286 y Fi({)s FG(2)1447 1269 y Fh(f)1445 1286 y FF(X)1540 1250 y Fr(\033)l Fy(\()p Ft(H)1709 1264 y Fi({)1763 1250 y Fy(\))h(and)g(the)g(fact)h(that)g(the)f(righ)m(t)g (hand)f(side)g(is)448 1363 y(closed)g(are)h(not)f(di\016cult)e(to)j (sho)m(w.)55 b(It)35 b(is)g(also)g(clear)g(that)h(in)d(this)i (generalit)m(y)g(one)448 1476 y(cannot)c(ha)m(v)m(e)h(equalit)m(y)1272 1443 y Fq(\(a\))1366 1476 y Fy(.)41 b(But)30 b(if)f Ft(H)38 b Fy(is)29 b(a\016liated)h(to)h Ft(C)2457 1443 y Fq(u)2450 1504 y(b)2500 1476 y Fy(\()p Ft(X)7 b Fy(\))21 b Fj(o)f Ft(X)38 b Fy(then)30 b(one)h(has)448 1680 y(\(4.5\))824 b Fr(\033)1511 1694 y Fq(ess)1602 1680 y Fy(\()p Ft(H)7 b Fy(\))26 b(=)1877 1612 y Fm(S)1953 1716 y Fi({)s FG(2)2052 1699 y Fh(f)2050 1716 y FF(X)2144 1680 y Fr(\033)l Fy(\()p Ft(H)2313 1694 y Fi({)2368 1680 y Fy(\))p Ft(:)448 1911 y Fy(The)f(description)f(of)1193 1888 y Fm(f)1190 1911 y Ft(X)48 b Fy(in)25 b(terms)g(of)h Ft(z)t Fy(-ultra\014lters)e(\(whic) m(h)h(can)h(b)s(e)f(found)g(in)f([GiJe])448 2024 y(or)31 b([W)-8 b(al)q(]\))30 b(sho)m(ws)h(that)f(this)g(is)f(a)i(strengthened) f(v)m(ersion)g(of)g(the)h(form)m(ula)f(\(4.4\).)589 2136 y(Our)k(pro)s(of)g(of)g(the)h(equalit)m(y)f(\(4.5\))j(is)c(based)i(on)f (a)h(certain)f(description)f(of)i(the)448 2249 y(quotien)m(t)28 b Ft(C)876 2216 y Fq(u)869 2277 y(b)919 2249 y Fy(\()p Ft(X)7 b Fy(\))p Ft(=C)1181 2263 y Fq(0)1222 2249 y Fy(\()p Ft(X)g Fy(\),)29 b(whic)m(h)d(w)m(e)i(presen)m(t)f(b)s(elo)m(w.)39 b(By)28 b(the)f(univ)m(ersal)f(prop)s(ert)m(y)448 2362 y(of)42 b Ft(\014)5 b(X)50 b Fy(eac)m(h)43 b Ft(')i Fl(2)f Ft(C)1241 2329 y Fq(u)1234 2390 y(b)1284 2362 y Fy(\()p Ft(X)7 b Fy(\))43 b(extends)f(to)h(an)e(elemen)m(t)i(of)f Ft(C)7 b Fy(\()p Ft(\014)e(X)i Fy(\))43 b(and)e(this)g(giv)m(es)448 2475 y(an)d(iden)m(ti\014cation)e(of)h(the)h(algebras)f Ft(C)1844 2442 y Fq(u)1837 2503 y(b)1887 2475 y Fy(\()p Ft(X)7 b Fy(\))39 b(and)d Ft(C)7 b Fy(\()p Ft(\014)e(X)i Fy(\).)63 b(The)37 b(restriction)f(map)448 2602 y Ft(')26 b Fl(7!)f Ft(')p Fl(j)735 2621 y Fh(f)733 2638 y FF(X)839 2602 y Fy(induces)g(an)h(isomorphism)e(b)s(et)m(w)m(een)j Ft(C)2228 2569 y Fq(u)2221 2630 y(b)2271 2602 y Fy(\()p Ft(X)7 b Fy(\))p Ft(=C)2533 2616 y Fq(0)2574 2602 y Fy(\()p Ft(X)g Fy(\))27 b(and)f Ft(C)7 b Fy(\()3036 2579 y Fm(f)3033 2602 y Ft(X)22 b Fy(\).)40 b(Ho)m(w-)448 2715 y(ev)m(er,)49 b(this)43 b(fact)j(do)s(es)e(not)g(seem)h(us)f(directly)f(relev)-5 b(an)m(t)44 b(for)g(the)h(pro)s(of)e(of)i(\(4.5\).)448 2828 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Fy(\))22 b Fj(o)e Ft(X)37 b Fy(whic)m(h,)30 b(in)f(turn,)g(allo)m(ws)h(one)h(to)g(pro)m(v)m(e)g(\(4.5\).)448 877 y Fz(4.3.)41 b Fy(Instead)29 b(of)g(considering)f(explicitly)e(the) k(Stone-)2358 854 y(\024)2348 877 y(Cec)m(h)g(compacti\014cation)f(of)h Ft(X)448 990 y Fy(one)37 b(can)f(construct)h(algebras)f Fn(A)58 b Fy(with)34 b(the)j(prop)s(erties)e(\(4.1\))j(b)m(y)e(using)e (limits)g(at)448 1103 y(in\014nit)m(y)27 b(along)i(certain)g (\014lters.)39 b(Although)28 b(a)i(\014lter)e(can)h(b)s(e)f(iden)m (ti\014ed)g(with)f(the)i(set)448 1216 y(of)d(ultra\014lters)e(\014ner)h (than)h(it,)g(this)f(p)s(oin)m(t)g(of)h(view)f(is)g(more)h(fruitful)d (in)i(applications,)448 1329 y(allo)m(wing)k(one)i(to)g(get)h(less)d (abstract)j(results.)589 1442 y(Let)40 b(us)e(sa)m(y)h(that)g(a)g (\014lter)f Fl(F)48 b Fy(is)37 b(a)i Ft(T)13 b(I)7 b(F)52 b Fy(if)38 b(it)g(is)f(translation)h(in)m(v)-5 b(arian)m(t)38 b(\()p Ft(A)i Fl(2)448 1554 y(F)9 b Ft(;)15 b(x)26 b Fl(2)f Ft(X)33 b Fl(\))25 b Ft(x)12 b Fy(+)g Ft(A)25 b Fl(2)g(F)9 b Fy(\))27 b(and)f(\014ner)f(than)h(the)h(F)-8 b(r)m(\023)-43 b(ec)m(het)29 b(\014lter.)39 b(T)-8 b(o)26 b(eac)m(h)i(suc)m(h)e(\014lter)448 1667 y(w)m(e)h(asso)s(ciate)g(the)f (space)h(of)f Ft(')g Fl(2)f Ft(C)1689 1634 y Fq(u)1682 1695 y(b)1732 1667 y Fy(\()p Ft(X)7 b Fy(\))27 b(whic)m(h)e(ha)m(v)m(e) i(a)g(limit)d(at)j(in\014nit)m(y)d(along)i(the)448 1780 y(\014lter:)448 1984 y(\(4.6\))568 b Ft(C)1266 1998 y FG(F)1327 1984 y Fy(\()p Ft(X)7 b Fy(\))27 b(=)e Fl(f)p Ft(')h Fl(2)f Ft(C)1890 1947 y Fq(u)1883 2008 y(b)1933 1984 y Fy(\()p Ft(X)7 b Fy(\))26 b Fl(j)g Fy(lim)2196 2045 y FG(F)2303 1984 y Ft(')31 b Fy(exists)o Fl(g)p Ft(:)448 2222 y Fy(It)d(is)f(clear)h(that)h Ft(C)1108 2236 y FG(F)1169 2222 y Fy(\()p Ft(X)7 b Fy(\))29 b(is)e(a)i(unital)d Ft(C)1845 2189 y FG(\003)1884 2222 y Fy(-subalgebra)h(of)h Ft(C)2540 2189 y Fq(u)2533 2250 y(b)2583 2222 y Fy(\()p Ft(X)7 b Fy(\))29 b(suc)m(h)f(that)g(\(4.1\))i(is)448 2335 y(satis\014ed.)39 b(If)26 b Fl(F)36 b 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Fy(-algebra.)47 b(It)33 b(will)d(also)i(b)s (e)g(con)m(v)m(enien)m(t)h(to)h(denote)e(b)m(y)h Ft(L)p Fy(-lim)13 b Ft(')33 b Fy(the)g(limit)d(along)448 4254 y(the)d(\014lter)e Fl(F)879 4268 y FF(L)932 4254 y Fy(.)39 b(Th)m(us,)27 b(if)e Ft(')h Fy(:)f Ft(X)33 b Fl(!)25 b Fp(C)50 b Fy(then)26 b Ft(L)p Fy(-lim)14 b Ft(')27 b Fy(exists)f(if)f(and)h(only)f(if)h(there)g(is)g(a)448 4367 y(complex)h(n)m(um)m(b)s(er)e Ft(c)i Fy(with)e(the)i(prop)s(ert)m (y:)39 b(for)26 b(eac)m(h)i Ft(")e(>)e Fy(0)k(one)f(can)g(\014nd)e(a)i (compact)448 4480 y(set)k(\003)26 b Fl(\032)f Ft(X)38 b Fy(suc)m(h)30 b(that)i Fl(j)p Ft(')p Fy(\()p Ft(x)p Fy(\))21 b Fl(\000)f Ft(c)p Fl(j)26 b Ft(<)g(")k Fy(if)g Ft(x)36 b(=)-56 b Fl(2)26 b Ft(L)2178 4494 y Fq(\003)2231 4480 y Fy(.)41 b(Then)29 b(lim)2660 4494 y FG(F)2710 4505 y Fv(L)2776 4480 y Ft(')d Fl(\021)g Ft(L)p Fy(-lim)13 b Ft(')26 b Fy(=)g Ft(c)p Fy(.)448 4593 y(No)m(w)40 b Ft(C)729 4607 y FF(L)781 4593 y Fy(\()p Ft(X)7 b Fy(\))40 b(is)e(the)h Ft(C)1310 4560 y 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b(classes)f(of)g(sets)g Ft(L)g Fy(or,)h(more)f (generally)-8 b(,)34 b Ft(T)13 b(I)7 b(F)13 b Fy('s,)34 b(for)e(whic)m(h)g(suc)m(h)h(\\explicit")448 1476 y(computations)27 b(can)h(b)s(e)e(done.)40 b(The)26 b(di\016cult)m(y)g(of)h(the)h (problem)d(app)s(ears)i(already)f(in)448 1589 y(the)31 b(seemingly)e(elemen)m(tary)i(case)g(when)f Ft(L)g Fy(is)f(a)i(straigh) m(t)g(line)d(in)h Ft(X)k Fy(=)25 b Fp(R)3054 1556 y Fq(2)3100 1589 y Fy(.)589 1702 y(W)-8 b(e)34 b(ha)m(v)m(e)f(considered)f(a)g (natural)g(generalization)g(of)g(the)h(preceding)e(situation.)448 1815 y(W)-8 b(e)29 b(do)f(not)g(giv)m(e)h(the)f(details)f(b)s(ecause)h (it)f(do)s(es)h(not)g(in)m(v)m(olv)m(e)g(essen)m(tially)f(new)g(ideas;) 448 1928 y(w)m(e)f(shall,)f(ho)m(w)m(ev)m(er,)k(describ)s(e)24 b(it)h(here)g(succinctly)-8 b(.)39 b(Let)26 b Ft(L)f Fy(b)s(e)g(the)h(union)e(of)i(a)g(family)448 2041 y Fl(B)k Fy(of)d(pairwise)e(disjoin)m(t)g(compact)j(sets)f(suc)m(h)g(that)g(for) g(eac)m(h)h(compact)g(\003)f(of)h Ft(X)34 b Fy(there)448 2154 y(is)23 b(a)i(\014nite)e(set)i Fl(B)1025 2168 y Fq(\003)1103 2154 y Fl(\032)g(B)h Fy(suc)m(h)e(that)h(\()p Ft(B)12 b Fy(+)c(\003\))g Fl(\\)g Fy(\()p Ft(B)2154 2121 y FG(0)2184 2154 y Fy(+)g(\003\))25 b(=)g Fl(;)f Fy(if)f Ft(B)40 b(=)-55 b Fl(2)25 b(B)2873 2168 y Fq(\003)2950 2154 y Fy(and)e Ft(B)3194 2121 y FG(0)3242 2154 y Fl(6)p Fy(=)i Ft(B)5 b Fy(.)448 2267 y(Then)27 b(w)m(e)i(sa)m(y)f(that)h Ft(L)f Fy(is)f(a)h Fo(disp)-5 b(erse)g(d)30 b Fy(set.)41 b(If)27 b(there)h(is)f(a)i(compact)g(set)f Ft(K)35 b Fy(suc)m(h)27 b(that)448 2379 y(eac)m(h)k Ft(B)f Fl(2)24 b(B)32 b Fy(is)d(a)g(subset)g(of)h(a)f(translate)h(of)f Ft(K)36 b Fy(\(i.e.)30 b Ft(L)f Fy(is)g(\\uniformly")e(disp)s(ersed\),) 448 2492 y(then)e Ft(L)f Fy(is)g(equiv)-5 b(alen)m(t)24 b(to)i(a)f(sparse)f(set,)j(in)c(the)i(sense)g(that)g Fl(F)2590 2506 y FF(L)2667 2492 y Fy(coincides)f(to)i Fl(F)3217 2512 y FF(L)3265 2493 y Fk(0)3328 2492 y Fy(for)448 2605 y(some)h(sparse)g(set)g Ft(L)1143 2572 y Fq(0)1182 2605 y Fy(.)39 b(Indeed,)27 b(it)f(su\016ces)h(to)g(replace)g(eac)m(h)g (compact)h(set)f Ft(B)j Fl(2)25 b(B)k Fy(b)m(y)448 2718 y(a)f(p)s(oin)m(t)e(sitting)h(inside)e(it.)39 b(As)27 b(w)m(e)h(shall)e(see,)j(in)d(the)h(case)i(of)e(a)h(sparse)f(set)h(the) f(main)448 2831 y(role)36 b(in)g(the)g(computation)h(of)f(the)h (quotien)m(t)g(is)e(pla)m(y)m(ed)i(b)m(y)f(the)h(algebra)f Ft(C)3153 2845 y Fq(0)3193 2831 y Fy(\()p Ft(X)7 b Fy(\))3345 2798 y Fq([)p FF(L)p Fq(])448 2944 y Fy(consisting)23 b(of)h(relativ)m(ely)g(compact)h(families)d Fl(f)p Ft(')2136 2959 y FF(l)2163 2944 y Fl(g)2208 2959 y FF(l)q FG(2)p FF(L)2354 2944 y Fy(of)i(elemen)m(ts)g(of)h Ft(C)2979 2958 y Fq(0)3018 2944 y Fy(\()p Ft(X)7 b Fy(\).)40 b(F)-8 b(or)25 b(a)448 3057 y(general)e(disp)s(ersed)d(set)j(this)e(has)i(to)g (b)s(e)f(replaced)g(with)f(the)i(algebra)g Ft(C)2896 3071 y Fq(0)2935 3057 y Fy(\()p Ft(X)7 b Fy(\))3087 3024 y Fq([)p FF(L;X)e Fq(])3285 3057 y Fy(\(see)448 3170 y Fl(x)p Fy(3.3)40 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b(suc)m(h)g(that)h Ft(\025)p Fy(\()p Ft(A)12 b Fl(\\)g Ft(B)5 b Fy(\))25 b Fl(\025)g Ft(\027)6 b(\025)p Fy(\()p Ft(B)f Fy(\))27 b(for)f(eac)m(h)h(ball)e Ft(B)30 b Fy(of)d(radius)d Ft(r)k Fl(\025)d Ft(r)s Fy(\()p Ft(\027)6 b Fy(\).)448 799 y(An)31 b(arbitrary)g(set)h(con)m(taining)f(a)h(Borel) f(full)f(set)i(will)d(also)i(b)s(e)g(called)g(full.)42 b(Clearly)-8 b(,)448 912 y(the)35 b(family)d(of)i(full)e(sets)i(is)f(a) i Ft(T)13 b(I)7 b(F)13 b Fy(;)36 b(w)m(e)e(denote)h(it)e Fl(L)p Fy(\()p Ft(X)7 b Fy(\))35 b(and)e(call)h(it)f(the)h Fo(L)-5 b(or)g(entz)448 1024 y(\014lter)31 b Fy(on)f Ft(X)7 b Fy(.)41 b(The)29 b(limit)f(along)i(the)g(\014lter)f Fl(L)p Fy(\()p Ft(X)7 b Fy(\))31 b(will)c(b)s(e)i(called)g Fo(L)-5 b(or)g(entz)34 b(limit)d Fy(and)448 1137 y(will)d(b)s(e)i (denoted)g Fs(Lim)16 b Ft(')31 b Fy(or)f Fs(Lim)1660 1151 y FF(x)p FG(!1)1845 1137 y Ft(')p Fy(\()p Ft(x)p Fy(\).)589 1250 y(F)-8 b(or)36 b(example,)g(assume)f(that)h Ft(A)1733 1217 y FF(c)1800 1250 y Fy(=)d Ft(X)16 b Fl(n)8 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y(one)k(has)f(to)h(consider)f(ab)s(elian)e (limits)g(instead)i(of)g(limits)e(in)h(Loren)m(tz)j(sense\).)589 1928 y(The)25 b Ft(C)843 1895 y FG(\003)882 1928 y Fy(-algebra)g Ft(C)1290 1942 y FG(L)1343 1928 y Fy(\()p Ft(X)7 b Fy(\))27 b(=)d Ft(C)1682 1946 y FG(L)p Fq(\()p FF(X)5 b Fq(\))1853 1928 y Fy(\()p Ft(X)i Fy(\))26 b(asso)s(ciated)g(to)g Fl(L)p Fy(\()p Ft(X)7 b Fy(\))26 b(is)e(called)g Fo(L)-5 b(or)g(entz)448 2041 y(algebr)g(a)32 b Fy(of)f Ft(X)7 b Fy(.)43 b(There)30 b(is)g(an)h(in)m(teresting)f(description)f(of)i (the)g(elemen)m(ts)g(of)g Ft(C)3206 2055 y FG(L)3259 2041 y Fy(\()p Ft(X)7 b Fy(\).)448 2154 y(By)40 b(de\014nition,)g(this) e(algebra)i(consists)f(of)g(all)g(functions)f Ft(')j Fl(2)f Ft(C)2821 2121 y Fq(u)2814 2181 y(b)2864 2154 y Fy(\()p Ft(X)7 b Fy(\))40 b(suc)m(h)f(that)448 2267 y Fs(Lim)634 2281 y FF(x)p FG(!1)819 2267 y Ft(')p Fy(\()p Ft(x)p Fy(\))30 b(exists.)40 b(But)30 b(a)f(b)s(ounded)e(Borel)j (function)d Ft(')j Fy(con)m(v)m(erges)h(in)d(Loren)m(tz)448 2379 y(sense)j(to)g(some)g(n)m(um)m(b)s(er)e Ft(c)h Fy(if)g(and)f(only) h(if)448 2595 y(\(4.8\))517 b(lim)1126 2649 y FF(r)r FG(!1)1322 2595 y Fy(sup)1316 2673 y FF(x)p FG(2)p FF(X)1549 2533 y Fy(1)p 1491 2574 162 4 v 1491 2657 a Ft(v)s Fy(\()p Ft(r)s Fy(\))1677 2471 y Fm(Z)1728 2677 y FF(B)s Fq(\()p FF(x)p Fq(;)p FF(r)r Fq(\))1952 2595 y Fl(j)p Ft(')p Fy(\()p Ft(y)s Fy(\))21 b Fl(\000)f Ft(c)p Fl(j)15 b Ft(\025)p Fy(\(d)q Ft(y)s Fy(\))25 b(=)g(0)p Ft(:)448 2831 y Fy(This)42 b(mak)m(es)i(the)g(connection)g(with)e(the)h(notion)g (of)h(almost)f(con)m(v)m(ergence)j(in)m(tro-)448 2944 y(duced)34 b(b)m(y)h(G.)g(C.)g(Loren)m(tz)h(for)e(the)h(case)h(of)f (sequences)g(and)g(extended)f(to)i(general)448 3057 y(semigroups)c(b)m (y)i(sev)m(eral)f(other)h(p)s(eople)e(\(see)j([P)m(at)q(])f(and)f (references)g(therein\).)50 b(The)448 3170 y(\\absolute")31 b(almost)f(con)m(v)m(ergence)i(that)e(w)m(e)h(denoted)f Fs(Lim)45 b Fy(ab)s(o)m(v)m(e)31 b(w)m(as)f(studied)f(es-)448 3283 y(p)s(ecially)f(b)m(y)i(C.)f(Chou,)g(see)i([Cho])e(and)h(also)f ([P)m(at)r(])h(for)f(additional)f(references.)41 b(The)448 3396 y(notion)28 b(of)h(\\full)d(set")k(and)e(the)g(presen)m(tation)h (in)e(terms)h(of)h(\014lters)e(do)s(es)h(not)h(seem)f(to)448 3509 y(b)s(e)j(standard,)h(but)e(is)h(quite)g(natural)g(and)g(it)g(is)f (easy)j(to)f(deduce)f(the)h(c)m(haracteriza-)448 3621 y(tion)i(\(4.8\))i(from)e(the)g(results)f(of)i(Chou)e(as)h(presen)m (ted)h(in)e([P)m(at)q(])h(\(see)i([GI1])f(for)f(the)448 3734 y(case)e Ft(X)g Fy(=)25 b Fp(R)s Fy(\))q(.)589 3847 y(By)h(an)f(observ)-5 b(ation)24 b(made)h(ab)s(o)m(v)m(e,)j(w)m(e)d(ha) m(v)m(e)h Ft(C)2238 3861 y FF(L)2290 3847 y Fy(\()p Ft(X)7 b Fy(\))27 b Fl(\032)e Ft(C)2630 3861 y FG(L)2683 3847 y Fy(\()p Ft(X)7 b Fy(\))26 b(for)e(eac)m(h)i(sparse)448 3960 y(set)34 b Ft(L)p Fy(.)47 b(Th)m(us)32 b(an)g(\\explicit")g (description)f(of)i(the)g(quotien)m(t)g Ft(C)2661 3974 y FG(L)2713 3960 y Fy(\()p Ft(X)7 b Fy(\))p Ft(=C)2975 3974 y Fq(0)3016 3960 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y Fq(0)1683 2999 y Fy(\()p Fp(R)1778 2961 y FG(\003)1824 2999 y Fy(\))20 b Fl(\010)g Ft(C)2042 2961 y Fb(E)2035 3021 y Fq(0)2089 2999 y Fy(\()p Fp(R)2185 2961 y FG(\003)2230 2999 y Fy(\))p Ft(:)448 3203 y Fy(Hence)27 b(if)d Ft(H)32 b Fy(is)25 b(an)g(observ)-5 b(able)25 b(a\016liated)g(to)h Fs(B)2109 3170 y Fb(E)2182 3203 y Fy(then)g(one)f(can)h(asso)s(ciate)h(to)f(it)f (four)448 3316 y(asymptotic)33 b(observ)-5 b(ables)32 b Ft(H)1475 3330 y FG(\006)1534 3316 y Ft(;)15 b(H)1657 3283 y FG(\006)1749 3316 y Fy(\(the)33 b(\014rst)g(t)m(w)m(o)h Ft(H)2384 3330 y FG(\006)2475 3316 y Fy(corresp)s(ond)e(to)h Ft(Q)c Fl(!)g(\0061)448 3429 y Fy(while)k(the)h(other)h(ones)f Ft(H)1381 3396 y FG(\006)1474 3429 y Fy(corresp)s(ond)f(to)j Ft(P)44 b Fl(!)32 b(\0061)p Fy(\))i(and)g(sp)s(ectral)g(prop)s(erties) 448 3542 y(lik)m(e)28 b(essen)m(tial)g(sp)s(ectrum,)g(thresholds,)f (Mourre)i(estimate,)g(of)g Ft(H)35 b Fy(can)29 b(b)s(e)f(expressed)448 3655 y(in)h(terms)i(of)f(these)h(observ)-5 b(ables.)40 b(F)-8 b(or)31 b(example)1034 3859 y Fr(\033)1088 3873 y Fq(ess)1179 3859 y Fy(\()p Ft(H)7 b Fy(\))26 b(=)f Fr(\033)l Fy(\()p Ft(H)1623 3873 y FG(\000)1682 3859 y Fy(\))c Fl([)f Fr(\033)l Fy(\()p Ft(H)1988 3873 y Fq(+)2047 3859 y Fy(\))g Fl([)g Fr(\033)l Fy(\()p Ft(H)2359 3821 y FG(\000)2419 3859 y Fy(\))g Fl([)g Fr(\033)l Fy(\()p Ft(H)2731 3821 y FG(\000)2790 3859 y Fy(\))p Ft(:)448 4063 y Fy(This)25 b(is)g(an)h(elemen)m(tary)g(but)g(ph)m(ysically)e(in) m(teresting)h(situation)g(when)g(the)h(essen)m(tial)448 4176 y(sp)s(ectrum)e(of)h Ft(H)32 b Fy(is)24 b(not)h(determined)e(b)m (y)i(its)f(lo)s(calizations)g(at)i(in\014nit)m(y)c(if)i(the)h (in\014nit)m(y)448 4289 y(is)30 b(in)m(terpreted)f(only)h(in)f(the)i Ft(Q)f Fy(sense.)589 4402 y(Detailed)g(pro)s(ofs)f(of)h(the)f (preceding)g(statemen)m(ts)i(and)e(of)h(the)g(Mourre)f(estimate)448 4515 y(for)34 b(op)s(erators)h(a\016liated)e(to)i Fs(C)1551 4482 y Fb(E)1633 4515 y Fy(or)f Fs(B)1836 4482 y Fb(E)1918 4515 y Fy(will)d(b)s(e)j(presen)m(ted)g(in)f([GI4)q(].)52 b(The)34 b(appli-)448 4628 y(cations)c(co)m(v)m(er)i(the)d(sp)s(ectral) g(theory)h(of)g(elliptic)e(op)s(erators)h(\(with)g(op)s(erator)h(v)-5 b(alued)448 4741 y(co)s(e\016cien)m(ts\))32 b(on)f(asymptotically)f (cylindric)e(\(star)j(shap)s(ed\))f(domains)g(in)f Fp(R)3141 4708 y FF(n)3225 4741 y Fy(\(with)448 4853 y(di\013eren)m(t)39 b(asymptotics)h(at)g(v)-5 b(arious)39 b(ends\))g(and)g(on)g(some)h (manifolds)d(with)h(cylin-)448 4966 y(drical)33 b(ends)h(\(our)g(w)m (ork)g(on)g(these)h(questions)e(has)h(b)s(een)g(esp)s(ecially)e(motiv) -5 b(ated)35 b(b)m(y)1897 5225 y(37)p eop %%Page: 38 38 38 37 bop 448 573 a Fy([Be)q(,)30 b(CrIf)o(,)f(DDI1)q(,)h(DDI2)q(])f (and)g(references)h(therein\).)39 b(By)30 b(taking)f Fg(E)c Fy(=)g Fp(C)3025 540 y Fq(2)3100 573 y Fy(w)m(e)k(co)m(v)m(er) 448 686 y(one)g(dimensional)d(Dirac)j(op)s(erators)g(with)f(di\013eren) m(t)g(asymptotics)h(at)h Fl(\0061)e Fy(\(in)g(fact,)448 799 y(b)m(y)22 b(taking)g(an)g(arbitrary)e Fg(E)i Fy(in)f(this)f(con)m (text)k(one)e(gets)h(a)f(generalized)g(Dirac)g(op)s(erator)448 912 y(whic)m(h)29 b(is)h(also)g(co)m(v)m(ered)i(b)m(y)e(the)h (formalism\).)589 1024 y(Finally)-8 b(,)26 b(note)i(that)f Fp(R)35 b Fy(could)26 b(b)s(e)f(replaced)h(b)m(y)h(an)f(arbitrary)f (\014nite)h(dimensional)448 1137 y(v)m(ector)38 b(space)e Ft(X)43 b Fy(and)p 1275 1064 66 4 v 35 w Fp(R)i Fy(b)m(y)36 b(the)f(spherical)f(compacti\014cation)p 2743 1064 83 4 v 37 w Ft(X)43 b Fy(de\014ned)34 b(b)m(y)i(the)448 1250 y(algebra)f Ft(C)7 b Fy(\()p 878 1177 V Ft(X)g Fy(\))35 b(of)f(functions)f Ft(')f Fl(2)g Ft(C)1789 1217 y Fq(u)1782 1278 y(b)1832 1250 y Fy(\()p Ft(X)7 b Fy(\))35 b(suc)m(h)f(that)h(lim) 2555 1265 y FF(\025)p FG(!)p Fq(+)p FG(1)2812 1250 y Ft(')p Fy(\()p Ft(\025x)p Fy(\))g(exists)f(lo-)448 1363 y(cally)e(uniformly)d(in)i Ft(x)d Fl(2)g Ft(X)7 b Fy(.)47 b(It)32 b(is)f(easy)i(to)g(compute)g(the)f(quotien)m(t)g Ft(C)7 b Fy(\()p 3016 1290 V Ft(X)h Fy(\))p Ft(=C)3244 1377 y Fq(0)3284 1363 y Fy(\()p Ft(X)f Fy(\))448 1476 y(in)29 b(this)h(case)h(but)f(the)g(applications)f(are)i(less)e(in)m (teresting)h(at)h(the)g(ph)m(ysical)e(lev)m(el.)448 1667 y Fz(4.6.)119 b Fy(W)-8 b(e)38 b(consider)d(here)i(a)g(class)f(of)h Ft(C)1939 1634 y FG(\003)1978 1667 y Fy(-algebras)g(whic)m(h)e(app)s (ear)h(in)f(the)i(study)448 1780 y(of)42 b Ft(N)10 b Fy(-b)s(o)s(dy)41 b(systems)g(and)g(their)g(generalizations.)75 b(A)41 b(detailed)g(presen)m(tation)h(of)448 1893 y(this)34 b(sub)5 b(ject)34 b(including)d(references)k(to)h(previous)d(w)m(orks)h (can)h(b)s(e)f(found)f(in)h([ABG)q(,)448 2006 y(BoGe)r(])h(for)g(the)g (case)h(of)f(\014nite)f(semilattices)g(and)g([DaG1)r(,)h(DaG2)r(])g (for)g(the)g(case)h(of)448 2119 y(arbitrary)41 b(semilattices.)75 b(Besides)41 b(the)i(non-relativistic)d Ft(N)10 b Fy(-b)s(o)s(dy)40 b(hamiltonians,)448 2232 y(this)32 b(framew)m(ork)g(co)m(v)m(ers)i(the) f(disp)s(ersiv)m(e)d(case)k(\(see)f([Dam)q(,)g(DaG2)q(])g(and)f (references)448 2345 y(therein\))24 b(and)f(the)h(class)g(of)g (pluristrati\014ed)c(media)j(\014rst)g(considered)g(b)m(y)h(Dermenjian) 448 2458 y(and)44 b(Iftimie)e(in)h([DeIf)q(],)k(see)e([DaG2)r(].)81 b(Moreo)m(v)m(er,)50 b Ft(N)10 b Fy(-b)s(o)s(dy)43 b(hamiltonians)e (with)448 2571 y(constan)m(t)29 b(magnetic)e(\014elds)f(\(cf.)h([GeLa)r (]\))g(also)g(b)s(elong)f(to)i(this)e(framew)m(ork,)i(but)e(this)448 2684 y(question)k(will)e(b)s(e)h(discussed)g(in)g([GI2)q(].)589 2796 y(A)34 b(family)d Fl(f)p Fs(C)1077 2810 y FF(i)1106 2796 y Fl(g)1151 2810 y FF(i)p FG(2)p FF(I)1295 2796 y Fy(of)i(subalgebras)f(of)h(an)g(algebra)g Fs(C)g Fy(is)f Fo(line)-5 b(arly)36 b(indep)-5 b(endent)448 2909 y Fy(if)35 b(for)g(eac)m(h)h(family)e Fl(f)p Ft(S)1274 2923 y FF(i)1303 2909 y Fl(g)1348 2923 y FF(i)p FG(2)p FF(I)1494 2909 y Fy(suc)m(h)i(that)g Ft(S)1963 2923 y FF(i)2024 2909 y Fl(2)d Fs(C)2179 2923 y FF(i)2241 2909 y Fl(8)p Ft(i;)15 b(S)2419 2923 y FF(i)2481 2909 y Fl(6)p Fy(=)33 b(0)j(for)f(at)i(most)e (a)h(\014nite)448 3022 y(n)m(um)m(b)s(er)g(of)h Ft(i)h Fy(and)1148 2954 y Fm(P)1244 3049 y FF(i)p FG(2)p FF(I)1370 3022 y Ft(S)1426 3036 y FF(i)1490 3022 y Fy(=)f(0,)i(one)e(has)g Ft(S)2105 3036 y FF(i)2169 3022 y Fy(=)g(0)g(for)g(all)f Ft(i)h Fl(2)f Ft(I)7 b Fy(.)60 b(The)37 b(sums)f(of)448 3135 y(algebras)31 b(whic)m(h)e(app)s(ear)g(b)s(elo)m(w)h(are)h (understo)s(o)s(d)d(in)h(the)i(sense)f(of)h(linear)e(spaces.)589 3248 y(Let)j Fl(L)g Fy(b)s(e)e(a)i Fo(semilattic)-5 b(e)p Fy(,)33 b(i.e.)44 b(a)32 b(partially)e(ordered)h(set)h(in)e(whic)m(h)g (eac)m(h)j(pair)d(of)448 3361 y(elemen)m(ts)f Ft(a;)15 b(b)29 b Fy(has)f(a)g(lo)m(w)m(er)h(b)s(ound)d Ft(a)16 b Fl(^)g Ft(b)p Fy(.)40 b(F)-8 b(or)29 b Ft(a)d Fl(2)e(L)k Fy(w)m(e)h(set)g Fl(L)2712 3375 y FF(a)2779 3361 y Fy(=)c Fl(f)p Ft(b)g Fl(2)g(Lj)p Ft(b)g Fl(\025)g Ft(a)p Fl(g)p Fy(;)448 3474 y(this)30 b(is)f(also)h(a)h(semilattice.)589 3587 y(W)-8 b(e)31 b(sa)m(y)e(that)h(a)g Ft(C)1243 3554 y FG(\003)1281 3587 y Fy(-algebra)g Fs(C)f Fy(is)f Fl(L)p Fo(-gr)-5 b(ade)g(d)40 b Fy(if)28 b(a)h(linearly)e(indep)s(enden)m(t)g (family)448 3700 y Fl(f)p Fs(C)p Fy(\()p Ft(a)p Fy(\))p Fl(g)717 3714 y FF(a)p FG(2L)887 3700 y Fy(of)j Ft(C)1062 3667 y FG(\003)1101 3700 y Fy(-subalgebras)f(of)i Fs(C)f Fy(has)g(b)s(een)g(giv)m(en)g(suc)m(h)g(that:)448 3838 y(\(i\))56 b Fs(C)p Fy(\()p Ft(a)p Fy(\))21 b Fl(\001)f Fs(C)p Fy(\()p Ft(b)p Fy(\))26 b Fl(\032)f Fs(C)p Fy(\()p Ft(a)c Fl(^)f Ft(b)p Fy(\))30 b(for)g(all)g Ft(a;)15 b(b)25 b Fl(2)g(L)p Fy(;)448 3975 y(\(ii\))45 b(if)29 b Fl(E)34 b(\032)25 b(L)30 b Fy(is)f(\014nite)g(then)1499 3907 y Fm(P)1595 4002 y FF(a)p FG(2E)1742 3975 y Fs(C)p Fy(\()p Ft(a)p Fy(\))i(is)f(a)g(closed)h(subspace)e(of)i Fs(C)p Fy(;)448 4113 y(\(iii\))610 4045 y Fm(P)706 4140 y FF(a)p FG(2L)858 4113 y Fs(C)p Fy(\()p Ft(a)p Fy(\))g(is)f(dense)g (in)f Fs(C)p Fy(.)448 4276 y(F)-8 b(or)24 b(eac)m(h)f Fl(E)34 b(\032)25 b(L)d Fy(w)m(e)h(de\014ne)f Fs(C)p Fy(\()p Fl(E)8 b Fy(\))23 b(as)g(the)f(closure)g(of)h Fs(C)p Fy(\()p Fl(E)8 b Fy(\))2485 4243 y FG(\016)2551 4276 y Fy(=)2647 4208 y Fm(P)2743 4303 y FF(a)p FG(2E)2889 4276 y Fs(C)p Fy(\()p Ft(a)p Fy(\).)39 b(If)22 b Ft(a)k Fl(2)e(L)448 4389 y Fy(w)m(e)35 b(set)f Fs(C)792 4403 y FF(a)865 4389 y Fy(=)d Fs(C)p Fy(\()p Fl(L)1126 4403 y FF(a)1167 4389 y Fy(\).)52 b(It)34 b(is)f(clear)g(that)i Fs(C)1958 4403 y FF(a)2033 4389 y Fy(is)e(a)h Fl(L)2270 4403 y FF(a)2312 4389 y Fy(-graded)g Ft(C)2716 4356 y FG(\003)2755 4389 y Fy(-subalgebra)f(of)h Fs(C)p Fy(.)448 4502 y(There)i(is)f(a)i(natural)e(map)h Fl(P)1498 4469 y FG(\016)1491 4524 y FF(a)1573 4502 y Fy(:)f Fs(C)p Fy(\()p Fl(L)p Fy(\))1827 4469 y FG(\016)1901 4502 y Fl(!)g Fs(C)p Fy(\()p Fl(L)2186 4516 y FF(a)2228 4502 y Fy(\))2263 4469 y FG(\016)2339 4502 y Fy(de\014ned)g(b)m(y)h Fl(P)2860 4469 y FG(\016)2853 4524 y FF(a)2915 4434 y Fm(P)3011 4529 y FF(b)p FG(2L)3156 4502 y Ft(T)13 b Fy(\()p Ft(b)p Fy(\))35 b(=)448 4546 y Fm(P)544 4641 y FF(b)p FG(2L)670 4649 y Fv(a)727 4615 y Ft(T)13 b Fy(\()p Ft(b)p Fy(\))30 b(if)e Ft(T)13 b Fy(\()p Ft(b)p Fy(\))25 b Fl(2)g Fs(C)p Fy(\()p Ft(b)p Fy(\))30 b(and)e Ft(T)13 b Fy(\()p Ft(b)p Fy(\))26 b Fl(6)p Fy(=)f(0)k(only)g(for)f(a)i(\014nite)e(n)m(um) m(b)s(ers)f(of)j Ft(b)p Fy(.)40 b(This)448 4728 y(map)d(is)g(clearly)f (a)i(surjectiv)m(e)f(morphism)e(of)i Fl(\003)p Fy(-algebras.)62 b(It)38 b(can)f(b)s(e)g(sho)m(wn)f(that)448 4840 y(this)27 b(map)h(is)f(con)m(tin)m(uous,)h(so)g(it)g(extends)g(to)g(a)h (surjectiv)m(e)f(morphism)d Fl(P)2989 4854 y FF(a)3056 4840 y Fy(:)h Fs(C)f Fl(!)g Fs(C)3370 4854 y FF(a)3412 4840 y Fy(.)448 4953 y(Moreo)m(v)m(er,)33 b Fl(P)936 4967 y FF(a)1008 4953 y Fy(is)c(a)h(pro)5 b(jection)30 b(\(in)f(the)h(sense)g(of)g(linear)f(spaces\))i(and)e(its)g(k)m(ernel)h (is)1897 5225 y(38)p eop %%Page: 39 39 39 38 bop 448 573 a Fs(C)p Fy(\()p Fl(L)607 540 y FG(0)607 595 y FF(a)649 573 y Fy(\),)31 b(where)f Fl(L)1066 540 y FG(0)1066 595 y FF(a)1132 573 y Fy(=)25 b Fl(f)p Ft(b)h Fl(2)f(Lj)p Ft(b)g Fj(\003)g Ft(a)p Fl(g)p Fy(.)589 686 y(Assume)36 b(that)h Fl(L)f Fy(has)g(a)g(least)h(elemen)m(t)f(min)14 b Fl(L)p Fy(,)38 b(denote)e Fl(M)h Fy(the)f(set)h(of)f(atoms)448 799 y(of)h Fl(L)g Fy(\(i.e.)59 b(minimal)34 b(elemen)m(ts)j(of)g Fl(L)24 b(n)h(f)p Fy(min)14 b Fl(Lg)p Fy(\))37 b(and)g(assume)f(that)h Fl(L)g Fy(is)e(atomic)448 912 y(\(i.e.)42 b(eac)m(h)32 b Ft(a)26 b Fl(6)p Fy(=)g(min)13 b Fl(L)31 b Fy(is)f(minorated)g(b)m(y) g(an)h(atom\).)43 b(Observ)m(e)31 b(that)g Fs(C)p Fy(\(min)14 b Fl(L)p Fy(\))31 b(is)f(a)448 1024 y(closed)h(self-adjoin)m(t)f(ideal) f(in)h Fs(C)p Fy(,)h(so)g(the)f Ft(C)1966 992 y FG(\003)2005 1024 y Fy(-algebra)h Fs(C)p Ft(=)p Fs(C)p Fy(\(min)15 b Fl(L)p Fy(\))31 b(is)e(w)m(ell)h(de\014ned.)448 1137 y(The)g(imp)s(ortan)m(t)f(fact)h(is)f(that)i(one)f(can)g(explicitly)d (realize)j(this)f(algebra)h(as)g(follo)m(ws.)448 1250 y(There)g(is)g(a)g(natural)g(morphism)1237 1441 y Fs(C)25 b Fl(3)g Ft(T)38 b Fl(7\000)-16 b(!)26 b Fy(\()p Fl(P)1770 1455 y 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b(natural)f(order)g(relation)g(\(so)h Fl(L)27 b(\021)g Fp(G)19 b Fy(\()p Ft(X)8 b Fy(\))38 b(the)32 b(grassmannian)e(of)i Ft(X)7 b Fy(\),)33 b(then)e(the)448 4863 y(family)36 b Fl(f)p Fs(C)p Fy(\()p Ft(Y)21 b Fy(\))p Fl(g)1028 4877 y FF(Y)16 b FG(2L)1223 4863 y Fy(is)36 b(a)i Fl(L)p Fy(-grading)f(on)g Fs(C)p Fy(.)62 b(The)37 b(op)s(erators)h(a\016liated)f(to)h Fs(C)g Fy(are)448 4976 y(studied)29 b(in)g([DaG2)r(].)1897 5225 y(39)p eop %%Page: 40 40 40 39 bop 589 573 a Fy(The)33 b(second)g(example)g(will)d(b)s(e)j (studied)e(in)h(detail)g(in)g(the)h(second)g(part)g(of)g(this)448 686 y(pap)s(er)f([GI2)q(],)h(so)g(again)f(w)m(e)h(do)f(not)h(en)m(ter)g (in)m(to)g(details.)46 b(W)-8 b(e)33 b(shall)e(presen)m(t)i(it)f(in)f (a)448 799 y(w)m(a)m(y)38 b(whic)m(h)d(suggests)i(generalizations)g(to) g(nilp)s(oten)m(t)e(groups)h(more)h(general)f(than)448 912 y(the)31 b(Heisen)m(b)s(erg)f(group)g(\(whic)m(h)f(is)g(implicit)f (b)s(elo)m(w\).)589 1024 y(W)-8 b(e)48 b(consider)e(a)h(system)g (\(\004)p 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Fy(\))p Fo(,)33 b(is)g(dense)g(in)f Ft(C)2412 4701 y FF(L;)p Fq(0)2519 4687 y Fy(\()p Ft(X)7 b Fy(\))p Fo(.)448 4863 y Fz(Pro)s(of:)48 b Fy(\(i\))22 b(Clearly)g Ft(C)1270 4877 y FF(L;)p Fq(c)1373 4863 y Fy(\()p Ft(X)7 b Fy(\))24 b(is)e(a)h(self-adjoin)m(t)f(\(non-closed\))h (ideal)f(in)f Ft(C)3042 4877 y FF(L)3095 4863 y Fy(\()p Ft(X)7 b Fy(\).)39 b(W)-8 b(e)448 4976 y(shall)29 b(pro)m(v)m(e)j(its)e (densit)m(y)g(in)f Ft(C)1519 4990 y FF(L;)p Fq(0)1626 4976 y Fy(\()p Ft(X)7 b Fy(\).)42 b(Let)32 b(\003)e(b)s(e)g(a)h (compact)h(neigh)m(b)s(ourho)s(o)s(d)c(of)j(0)1897 5225 y(42)p eop %%Page: 43 43 43 42 bop 448 573 a Fy(in)30 b Ft(X)39 b Fy(and)30 b(let)h Ft(\022)e Fl(2)d Ft(C)1202 587 y Fq(c)1237 573 y Fy(\(\003\))32 b(suc)m(h)f(that)h(0)26 b Fl(\024)h Ft(\022)h Fl(\024)e Fy(1)32 b(and)e Ft(\022)f Fy(=)d(1)32 b(on)e(a)i(neigh)m(b)s(ourho)s(o) s(d)448 686 y(\003)511 700 y Fq(0)583 686 y Fy(of)g(zero.)47 b(Denote)34 b Ft(n)d Fy(the)h(maximal)f(n)m(um)m(b)s(er)g(of)h(sets)h Ft(l)23 b Fy(+)e(\003,)33 b(with)e Ft(l)f Fl(2)e Ft(L)p Fy(,)k(whic)m(h)448 799 y(ha)m(v)m(e)g(a)f(non-empt)m(y)g(in)m (tersection.)41 b(Since)29 b Ft(L)h Fy(is)g(a)h(sparse)f(subset)g(of)h Ft(X)38 b Fy(the)30 b(n)m(um)m(b)s(er)448 912 y Ft(n)35 b Fy(is)e(\014nite.)53 b(Then)34 b(\002\()p Ft(x)p Fy(\))f(=)1485 843 y Fm(P)1581 938 y FF(l)q FG(2)p FF(L)1717 912 y Ft(\022)s Fy(\()p Ft(x)23 b Fl(\000)f Ft(l)r Fy(\))35 b(is)f(w)m(ell)g (de\014ned,)h(0)d Fl(\024)h Fy(\002\()p Ft(x)p Fy(\))g Fl(\024)f Ft(n)i Fy(\(for)448 1024 y(eac)m(h)e Ft(x)e Fy(there)h(are)f(at)i(most)e Ft(n)g Fy(nonzero)h(terms)f(in)f(the)i (sum\))f(and)778 1222 y Fl(j)p Fy(\002\()p Ft(x)p Fy(\))21 b Fl(\000)f Fy(\002\()p Ft(y)s Fy(\))p Fl(j)25 b(\024)g Ft(n)15 b Fy(sup)1523 1301 y FF(l)q FG(2)p FF(L)1665 1222 y Fl(j)p Ft(\022)s Fy(\()p Ft(x)20 b Fl(\000)g Ft(l)r Fy(\))g Fl(\000)g Ft(\022)s Fy(\()p Ft(y)j Fl(\000)d Ft(l)r Fy(\))p Fl(j)26 b(\024)f Ft(n)p Fl(k)p Ft(\034)2700 1236 y FF(y)r FG(\000)p FF(x)2835 1222 y Ft(\022)e Fl(\000)d Ft(\022)s Fl(k)p Ft(;)448 1476 y Fy(where)27 b Fl(k)15 b(\001)g(k)29 b Fy(is)d(the)i(sup)e(norm.)39 b(So)28 b(\002)f(is)g(uniformly)d(con)m(tin)m(uous)k(and)f(\002)e Fl(2)g Ft(C)3156 1490 y FF(L;)p Fq(c)3259 1476 y Fy(\()p Ft(X)7 b Fy(\).)589 1589 y(Let)33 b Ft(')27 b Fl(2)g Ft(C)993 1603 y FF(L;)p Fq(0)1100 1589 y Fy(\()p Ft(X)7 b Fy(\))33 b(and)e Ft(")c(>)g Fy(0.)45 b(Then)30 b(there)i(is)f(a)h (compact)g(neigh)m(b)s(ourho)s(o)s(d)d(\003)448 1702 y(of)34 b(zero)h(in)e Ft(X)41 b Fy(suc)m(h)34 b(that)g Fl(j)p Ft(')p Fy(\()p Ft(x)p Fy(\))p Fl(j)f Ft(<)e(")j Fy(if)f Ft(x)41 b(=)-55 b Fl(2)31 b Ft(L)22 b Fy(+)h(\003.)51 b(Cho)s(ose)34 b Ft(F)44 b Fl(\032)31 b Ft(L)i Fy(\014nite)g(suc)m(h) 448 1814 y(that)c(\()p Ft(l)19 b Fy(+)d(\003\))g Fl(\\)g Fy(\()p Ft(l)1066 1781 y FG(0)1106 1814 y Fy(+)g(\003\))25 b(=)g Fl(;)k Fy(if)e Ft(l)h Fl(2)c Ft(M)36 b Fl(\021)25 b Ft(L)16 b Fl(n)g Ft(F)42 b Fy(and)28 b Ft(l)2370 1781 y FG(0)2418 1814 y Fl(2)d Ft(L;)40 b(l)2660 1781 y FG(0)2709 1814 y Fl(6)p Fy(=)25 b Ft(l)r Fy(,)k(and)e(denote)i Ft(K)448 1927 y Fy(the)i(compact)h(set)1112 1859 y Fm(S)1187 1954 y FF(l)q FG(2)p FF(F)1315 1927 y Fy([)p Ft(l)23 b Fy(+)d(\003].)42 b(Observ)m(e)31 b(that)g(if)f Ft(x)36 b(=)-55 b Fl(2)25 b Ft(K)37 b Fy(and)31 b Ft(\022)i Fy(is)c(as)i(ab)s (o)m(v)m(e)h(then)448 2040 y(\002\()p Ft(x)p Fy(\))g(=)774 1972 y Fm(P)870 2067 y FF(l)q FG(2)p FF(M)1033 2040 y Ft(\022)s Fy(\()p Ft(x)22 b Fl(\000)g Ft(l)r Fy(\))34 b(and)f(the)h(supp)s(orts)e(of)i(the)g(functions)e Ft(\034)2797 2055 y FF(l)2823 2040 y Ft(\022)k Fy(are)e(disjoin)m(t)e(if)448 2153 y Ft(l)27 b Fl(2)e Ft(M)10 b Fy(;)28 b(in)c(particular)g(0)i Fl(\024)f Fy(\002\()p Ft(x)p Fy(\))h Fl(\024)f Fy(1.)39 b(Let)26 b Ft(\021)j Fy(b)s(e)c(a)h(con)m(tin)m(uous)g(function)e(suc)m (h)h(that)448 2266 y(0)k Fl(\024)f Ft(\021)k Fl(\024)c Fy(1,)33 b Ft(\021)f Fy(=)c(0)33 b(on)f(a)g(neigh)m(b)s(orho)s(o)s(d)e (of)j Ft(K)38 b Fy(and)32 b Ft(\021)g Fy(=)c(1)k(on)g(a)h(neigh)m(b)s (orho)s(o)s(d)d(of)448 2379 y(in\014nit)m(y)-8 b(.)39 b(If)30 b(w)m(e)h(denote)g Ft(\021)s(')26 b Fy(=)f Ft( )34 b Fy(w)m(e)d(ha)m(v)m(e)1341 2577 y Ft(')21 b Fl(\000)f Fy(\(1)h Fl(\000)f Ft(\021)s Fy(\))p Ft(')h Fl(\000)f Fy(\002)p Ft( )28 b Fy(=)d Ft( )f Fl(\000)c Fy(\002)p Ft( )s(:)448 2774 y Fy(If)40 b Ft(x)h Fl(2)g Ft(K)46 b Fy(then)40 b(the)g(r.h.s.)g(ab)s(o)m(v)m(e)h(tak)m(es)g(the)f(v)-5 b(alue)40 b(zero)h(in)d Ft(x)p Fy(.)70 b(If)39 b Ft(x)51 b(=)-55 b Fl(2)41 b Ft(K)47 b Fy(and)448 2887 y Ft(x)e Fl(2)f Ft(L)27 b Fy(+)h(\003)901 2901 y Fq(0)982 2887 y Fy(then)42 b(there)g(is)f(a)h(unique)e Ft(l)47 b Fl(2)d Ft(M)52 b Fy(suc)m(h)41 b(that)i Ft(x)h Fl(2)g Ft(l)30 b Fy(+)d(\003)3105 2901 y Fq(0)3145 2887 y Fy(,)45 b(hence)448 3000 y(\002\()p Ft(x)p Fy(\))32 b(=)f Ft(\022)s Fy(\()p Ft(x)22 b Fl(\000)g Ft(l)r Fy(\))32 b(=)f(1)j(and)f Ft( )s Fy(\()p Ft(x)p Fy(\))24 b Fl(\000)e Fy(\002\()p Ft(x)p Fy(\))p Ft( )s Fy(\()p Ft(x)p Fy(\))33 b(=)e(0.)52 b(If)33 b Ft(x)41 b(=)-55 b Fl(2)31 b Ft(K)41 b Fy(and)33 b 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y(the)h(case)g(of)g(the)f(algebra)h Ft(C)1441 3822 y FF(L;)p Fq(0)1548 3808 y Fy(\()p Ft(X)7 b Fy(\))21 b Fj(o)f Ft(X)7 b Fy(.)448 3960 y Fz(Pro)s(of:)48 b Fy(The)31 b(uniqueness)e(of)i Fl(J)48 b Fy(is)30 b(a)i(consequence)g(of)g(\(i\))f (from)g(Lemma)h(5.1.)45 b(The)448 4073 y(surjectivit)m(y)39 b(of)h Fl(J)57 b Fy(is)39 b(also)h(easy)g(to)h(pro)m(v)m(e:)60 b(since)40 b(the)g(range)g(of)g(a)g(morphism)e(is)448 4186 y(closed,)28 b(it)e(su\016ces)g(to)h(sho)m(w)g(that)g(the)g(set)g (of)f(elemen)m(ts)h(of)g(the)g(form)f(\()p Ft(')2969 4201 y FF(l)2996 4186 y Fy(\))3031 4201 y FF(l)q FG(2)p FF(L)3152 4186 y Fy(,)i(where)448 4299 y(the)22 b(family)e Fl(f)p Ft(')969 4314 y FF(l)996 4299 y Fl(g)1041 4314 y FF(l)q FG(2)p FF(L)1184 4299 y Fy(is)h(as)h(in)e(the)i(statemen)m(t)i (of)e(the)f(theorem,)k(is)20 b(dense)i(in)e Ft(C)3128 4313 y Fq(0)3167 4299 y Fy(\()p Ft(X)7 b Fy(\))3319 4266 y Fq([)p FF(L)p Fq(])3412 4299 y Fy(.)448 4412 y(But)40 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(set,)48 b(the)c(functions)e Ft(')1901 4991 y FF(l)1971 4976 y Fy(whic)m(h)h(corresp)s(ond)f(to)j(large)f(enough)f Ft(l)1897 5225 y Fy(44)p eop %%Page: 45 45 45 44 bop 448 573 a Fy(are)40 b(uniquely)d(de\014ned.)67 b(In)38 b(particular,)i(the)g(image)g(of)f Fl(f)p Ft(')2595 588 y FF(l)2622 573 y Fl(g)2667 588 y FF(l)q FG(2)p FF(L)2828 573 y Fy(in)f(the)i(quotien)m(t)448 686 y Ft(C)513 700 y Fq(0)553 686 y Fy(\()p Ft(X)7 b Fy(\))705 653 y Fq([)p FF(L)p Fq(])797 686 y Ft(=C)907 700 y Fq(0)947 686 y Fy(\()p Ft(X)g Fy(\))1099 653 y Fq(\()p FF(L)p Fq(\))1239 686 y Fy(dep)s(ends)30 b(only)i(on)g Ft(')p Fy(.)46 b(Th)m(us)31 b Fl(J)48 b Fy(is)31 b(w)m(ell)g(de\014ned)g(on)h Ft(C)3181 700 y FF(L;)p Fq(c)3284 686 y Fy(\()p Ft(X)7 b Fy(\))448 799 y(and)32 b(it)g(is)g(clearly)g(a)g(morphism)f(with)g Fl(kJ)17 b(k)29 b(\024)f Fy(1.)48 b(This)31 b(allo)m(ws)g(us)h(to)i (extend)e(it)g(b)m(y)448 912 y(con)m(tin)m(uit)m(y)f(to)g(all)e Ft(C)1180 926 y FF(L;)p Fq(0)1287 912 y Fy(\()p Ft(X)7 b Fy(\).)589 1024 y(It)42 b(remains)e(to)j(pro)m(v)m(e)f(that)g(the)g (k)m(ernel)f(of)h Fl(J)57 b Fy(is)41 b Ft(C)2485 1038 y Fq(0)2524 1024 y Fy(\()p Ft(X)7 b Fy(\).)76 b(W)-8 b(e)42 b(\014rst)f(mak)m(e)i(a)448 1137 y(preliminary)31 b(general)j(remark.)52 b Fo(L)-5 b(et)37 b Fn(A)57 b Fo(b)-5 b(e)36 b(a)g Ft(C)2175 1104 y FG(\003)2214 1137 y Fo(-algebr)-5 b(a)37 b(and)g(let)g Fr(S)f Fy(=)31 b(\()p Ft(S)3162 1152 y FF(l)3188 1137 y Fy(\))3223 1152 y FF(l)q FG(2)p FF(L)3377 1137 y Fl(2)448 1250 y Fn(A)543 1217 y Fq([)p FF(L)p Fq(])675 1250 y Fo(such)42 b(that)g Fl(k)1147 1227 y Fm(b)1130 1250 y Fr(S)21 b Fl(k)42 b Ft(<)e(")i Fo(for)g(some)g Ft(")g(>)f Fy(0)p Fo(,)j(wher)-5 b(e)2485 1227 y Fm(b)2469 1250 y Fr(S)61 b Fo(is)42 b(the)g(image)f(of)h Fr(S)k Fo(in)448 1363 y Fn(A)543 1330 y FG(h)p FF(L)p FG(i)675 1363 y Fl(\021)25 b Fn(A)865 1330 y Fq([)p FF(L)p Fq(])956 1363 y Ft(=)p Fn(A)1096 1330 y Fq(\()p FF(L)p Fq(\))1203 1363 y Fo(.)42 b(Then)33 b(ther)-5 b(e)33 b(is)f(a)h(\014nite)g(set)f Ft(F)39 b Fl(\032)25 b Ft(L)32 b Fo(such)h(that)g Fl(k)p Ft(S)3068 1378 y FF(l)3094 1363 y Fl(k)26 b Ft(<)f Fy(2)p Ft(")33 b Fo(if)448 1476 y Ft(l)38 b(=)-56 b Fl(2)25 b Ft(F)13 b Fy(.)41 b(Indeed,)991 1679 y Fl(k)1052 1656 y Fm(b)1036 1679 y Fr(S)20 b Fl(k)26 b Fy(=)f(inf)6 b Fl(fk)p Fy(\()p Ft(S)1578 1694 y FF(l)1625 1679 y Fl(\000)20 b Ft(T)1769 1694 y FF(l)1795 1679 y Fy(\))1830 1694 y FF(l)q FG(2)p FF(L)1951 1679 y Fl(k)26 b(j)f Fr(T)40 b Fl(\021)25 b Fy(\()p Ft(T)2354 1694 y FF(l)2380 1679 y Fy(\))2415 1694 y FF(l)q FG(2)p FF(L)2562 1679 y Fl(2)g Fn(A)2742 1642 y Fq(\()p FF(L)p Fq(\))2849 1679 y Fl(g)448 1883 y Fy(so)36 b(there)g(is)f Fr(T)48 b Fl(2)34 b Fn(A)1195 1850 y Fq(\()p FF(L)p Fq(\))1338 1883 y Fy(suc)m(h)h(that)i Fl(k)p Fy(\()p Ft(S)1887 1898 y FF(l)1937 1883 y Fl(\000)23 b Ft(T)2084 1898 y FF(l)2110 1883 y Fy(\))2145 1898 y FF(l)q FG(2)p FF(L)2267 1883 y Fl(k)34 b Fy(=)g(sup)2588 1904 y FF(l)q FG(2)p FF(L)2724 1883 y Fl(k)p Ft(S)2825 1898 y FF(l)2875 1883 y Fl(\000)24 b Ft(T)3023 1898 y FF(l)3049 1883 y Fl(k)34 b Ft(<)g Fy(3)p Ft("=)p Fy(2.)448 1995 y(Then)g Fl(k)p Ft(S)791 2010 y FF(l)817 1995 y Fl(k)e Ft(<)f Fl(k)p Ft(T)1094 2010 y FF(l)1121 1995 y Fl(k)23 b Fy(+)g(3)p Ft("=)p Fy(2)36 b(and)d Fl(k)p Ft(T)1774 2010 y FF(l)1801 1995 y Fl(k)f(!)g Fy(0)j(as)f Ft(l)g Fl(!)d(1)p Fy(.)53 b(So)34 b(there)g(is)g(a)g(\014nite)f(set)448 2108 y Ft(F)39 b Fl(\032)25 b Ft(L)30 b Fy(suc)m(h)g(that)h Fl(k)p Ft(T)1233 2123 y FF(l)1259 2108 y Fl(k)26 b Ft(<)f("=)p Fy(2)32 b(if)d Ft(l)37 b(=)-55 b Fl(2)25 b Ft(F)13 b Fy(,)31 b(whic)m(h)e(pro)m(v)m(es)i(the)f(remark.)589 2221 y(Let)42 b Ft(')i Fl(2)f Ft(C)1035 2235 y FF(L;)p Fq(0)1142 2221 y Fy(\()p Ft(X)7 b Fy(\))42 b(suc)m(h)f(that)h Fl(J)16 b Fy(\()p Ft(')p Fy(\))44 b(=)f(0)f(and)e(let)h Ft(")j(>)f Fy(0.)73 b(Then)41 b(there)g(is)448 2334 y Ft( )d Fl(2)33 b Ft(C)704 2348 y FF(L;)p Fq(c)807 2334 y Fy(\()p Ft(X)7 b Fy(\))37 b(suc)m(h)e(that)h Fl(k)p Ft(')25 b Fl(\000)e Ft( )s Fl(k)35 b Ft(<)e(")j Fy(and)f Fl(kJ)17 b Fy(\()p Ft( )s Fy(\))p Fl(k)35 b Ft(<)f(")p Fy(.)57 b(Cho)s(ose)35 b(a)h(compact)448 2447 y(\003)i Fl(\032)f Ft(X)45 b Fy(and)37 b(a)h(b)s(ounded)e(equicon)m(tin)m(uous)g(family)g Fl(f)p Ft( )2438 2462 y FF(l)2465 2447 y Fl(g)2510 2462 y FF(l)q FG(2)p FF(L)2669 2447 y Fy(in)g Ft(C)2847 2461 y Fq(0)2887 2447 y Fy(\(\003\))i(suc)m(h)f(that)448 2560 y Ft( )k Fy(=)655 2492 y Fm(P)751 2587 y FF(l)q FG(2)p FF(L)887 2560 y Ft(\034)927 2575 y FF(l)953 2560 y Ft( )1012 2575 y FF(l)1038 2560 y Fy(.)61 b(Then)37 b(\()p Ft( )1463 2575 y FF(l)1489 2560 y Fy(\))1524 2575 y FF(l)q FG(2)p FF(L)1682 2560 y Fl(2)g Ft(C)1845 2574 y Fq(0)1884 2560 y Fy(\(\003\))2017 2527 y Fq([)p FF(L)p Fq(])2147 2560 y Fy(and)f Fl(k)p Ft(\031)s Fy([\()p Ft( )2549 2575 y FF(l)2577 2560 y Fy(\))2612 2575 y FF(l)q FG(2)p FF(L)2733 2560 y Fy(])p Fl(k)i Fy(=)e Fl(kJ)17 b Fy(\()p Ft( )s Fy(\))p Fl(k)38 b Ft(<)f(")448 2673 y Fy(hence,)28 b(b)m(y)e(the)h (preceding)e(remark,)i(there)g(is)e(a)i(\014nite)f(set)g Ft(F)39 b Fl(\032)25 b Ft(L)h Fy(suc)m(h)g(that)h Fl(k)p Ft( )3269 2688 y FF(l)3296 2673 y Fl(k)e Ft(<)448 2786 y Fy(2)p Ft(")36 b Fy(if)f Ft(l)45 b(=)-55 b Fl(2)33 b Ft(F)13 b Fy(.)56 b(But)36 b Ft( )s Fy(\()p Ft(x)p Fy(\))e(=)1477 2718 y Fm(P)1573 2813 y FF(l)q FG(2)p FF(L)1709 2786 y Ft( )1768 2801 y FF(l)1794 2786 y Fy(\()p Ft(x)24 b Fl(\000)f Ft(l)r Fy(\))36 b(and)e(if)h Ft(x)g Fy(is)f(outside)h(some)h(compact)448 2899 y(then)d(at)i(most)e(one)h (term)g(in)e(the)h(sum)g(is)f(non-zero,)j(so)f Fl(j)p Ft( )s Fy(\()p Ft(x)p Fy(\))p Fl(j)e Ft(<)e(")k Fy(for)f Ft(x)h Fy(in)e(some)448 3012 y(neigh)m(b)s(ourho)s(o)s(d)i(of)k (in\014nit)m(y)-8 b(.)58 b(Then)36 b Fl(j)p Ft(')p Fy(\()p Ft(x)p Fy(\))p Fl(j)i(\024)e(j)p Ft(')p Fy(\()p Ft(x)p Fy(\))26 b Fl(\000)e Ft( )s Fy(\()p Ft(x)p Fy(\))p Fl(j)i Fy(+)f Fl(j)p Ft( )s Fy(\()p Ft(x)p Fy(\))p Fl(j)38 b Ft(<)e Fy(3)p Ft(")h Fy(for)448 3125 y(suc)m(h)30 b Ft(x)p Fy(.)41 b(Since)29 b Ft(")i Fy(is)e(arbitrary)-8 b(,)30 b(this)g(sho)m(ws)g Ft(')25 b Fl(2)g Ft(C)2251 3139 y Fq(0)2291 3125 y Fy(\()p Ft(X)7 b Fy(\).)p 3371 3117 67 67 v 448 3385 a Fz(Corollary)35 b(5.3)46 b Fo(The)37 b(quotient)f(algebr)-5 b(a)38 b Ft(C)2013 3399 y FF(L)2065 3385 y Fy(\()p Ft(X)7 b Fy(\))p Ft(=C)2327 3399 y Fq(0)2368 3385 y Fy(\()p Ft(X)g Fy(\))37 b Fo(is)g(c)-5 b(anonic)g(al)5 b(ly)38 b(isomor-)448 3498 y(phic)33 b(to)h(the)e(unital)i Ft(C)1236 3465 y FG(\003)1275 3498 y Fo(-algebr)-5 b(a)33 b(asso)-5 b(ciate)g(d)35 b(to)e Ft(C)2214 3512 y Fq(0)2253 3498 y Fy(\()p Ft(X)7 b Fy(\))2405 3465 y FG(h)p FF(L)p FG(i)2514 3498 y Fo(.)41 b(In)33 b(p)-5 b(articular,)35 b(ther)-5 b(e)33 b(is)448 3611 y(a)g(natur)-5 b(al)35 b(emb)-5 b(e)g(dding:)448 3846 y Fy(\(5.2\))1153 3810 y Ft(C)1218 3824 y FF(L)1270 3810 y Fy(\()p Ft(X)7 b Fy(\))1408 3745 y Fm(.)1466 3889 y Ft(C)1531 3903 y Fq(0)1571 3889 y Fy(\()p Ft(X)g Fy(\))1782 3846 y Ft(,)-15 b Fl(!)1941 3810 y Ft(C)2006 3824 y FG(1)2080 3810 y Fy(\()p Ft(X)7 b Fy(\))2232 3777 y Fq([)p FF(L)p Fq(])2310 3745 y Fm(.)2368 3889 y Ft(C)2433 3903 y Fq(0)2473 3889 y Fy(\()p Ft(X)g Fy(\))2625 3856 y Fq(\()p FF(L)p Fq(\))448 4081 y Fz(Pro)s(of:)48 b Fy(W)-8 b(e)31 b(ha)m(v)m(e)448 4285 y Ft(C)513 4299 y FF(L)565 4285 y Fy(\()p Ft(X)7 b Fy(\))p Ft(=C)827 4299 y Fq(0)868 4285 y Fy(\()p Ft(X)g Fy(\))1105 4260 y Fl(\030)1105 4289 y Fy(=)1258 4285 y(\()p Fp(C)45 b Fy(+)20 b Ft(C)1536 4299 y FF(L;)p Fq(0)1643 4285 y Fy(\()p Ft(X)7 b Fy(\)\))p Ft(=C)1940 4299 y Fq(0)1981 4285 y Fy(\()p Ft(X)g Fy(\))2190 4260 y Fl(\030)2190 4289 y Fy(=)2316 4285 y Fp(C)44 b Fy(+)20 b Ft(C)2558 4299 y FF(L;)p Fq(0)2665 4285 y Fy(\()p Ft(X)7 b Fy(\))p Ft(=C)2927 4299 y Fq(0)2968 4285 y Fy(\()p Ft(X)g Fy(\))1105 4407 y Fl(\030)1105 4436 y Fy(=)1258 4432 y Fp(C)45 b Fy(+)19 b Ft(C)1500 4446 y Fq(0)1540 4432 y Fy(\()p Ft(X)7 b Fy(\))1692 4395 y Fq([)p FF(L)p Fq(])1784 4432 y Ft(=C)1894 4446 y Fq(0)1934 4432 y Fy(\()p Ft(X)g Fy(\))2086 4395 y Fq(\()p FF(L)p Fq(\))2250 4407 y Fl(\030)2250 4436 y Fy(=)2376 4432 y(\()p Fp(C)45 b Fy(+)20 b Ft(C)2654 4446 y Fq(0)2693 4432 y Fy(\()p Ft(X)7 b Fy(\))2845 4395 y Fq([)p FF(L)p Fq(])2938 4432 y Fy(\))p Ft(=C)3083 4446 y Fq(0)3123 4432 y Fy(\()p Ft(X)g Fy(\))3275 4395 y Fq(\()p FF(L)p Fq(\))3413 4432 y Ft(:)448 4635 y Fy(Moreo)m(v)m(er,)33 b(b)m(y)d(Lemma)h(2.3)g(one)g(has)484 4839 y Fp(C)44 b Fy(+)20 b Ft(C)726 4853 y Fq(0)765 4839 y Fy(\()p Ft(X)7 b Fy(\))917 4801 y Fq([)p FF(L)p Fq(])1108 4813 y Fl(\030)1108 4843 y Fy(=)1277 4839 y Fp(C)44 b Fy(+)20 b Ft(l)1483 4801 y FG(1)1558 4839 y Fy(\()p Ft(L)p Fy(\))g Fl(\012)g Ft(C)1866 4853 y Fq(0)1906 4839 y Fy(\()p Ft(X)7 b Fy(\))1093 4976 y Ft(,)-15 b Fl(!)83 b Ft(l)1306 4939 y FG(1)1381 4976 y Fy(\()p Ft(L)p Fy(\))20 b Fl(\012)g Fy([)p Fp(C)45 b Fy(+)20 b Ft(C)1892 4990 y Fq(0)1931 4976 y Fy(\()p Ft(X)7 b Fy(\)])27 b(=)e Ft(l)2260 4939 y FG(1)2334 4976 y Fy(\()p Ft(L)p Fy(\))c Fl(\012)f Ft(C)2643 4990 y FG(1)2718 4976 y Fy(\()p Ft(X)7 b Fy(\))2896 4951 y Fl(\030)2896 4980 y Fy(=)2992 4976 y Ft(C)3057 4990 y FG(1)3132 4976 y Fy(\()p Ft(X)g Fy(\))3284 4939 y Fq([)p FF(L)p Fq(])3376 4976 y Ft(:)1897 5225 y Fy(45)p eop %%Page: 46 46 46 45 bop 448 573 a Fy(In)35 b(other)g(terms,)i(the)f(canonical)f(em)m (b)s(edding)e Fp(C)48 b Fy(+)23 b Ft(C)2373 587 y Fq(0)2412 573 y Fy(\()p Ft(X)7 b Fy(\))2564 540 y Fq([)p FF(L)p Fq(])2690 573 y Ft(,)-15 b Fl(!)34 b Ft(C)2890 587 y FG(1)2964 573 y Fy(\()p Ft(X)7 b Fy(\))3116 540 y Fq([)p FF(L)p Fq(])3244 573 y Fy(asso-)448 686 y(ciates)31 b(to)g Ft(\025)21 b Fy(+)f(\()p Ft(')1071 701 y FF(l)1097 686 y Fy(\))1132 701 y FF(l)q FG(2)p FF(L)1284 686 y Fy(the)31 b(elemen)m(t)g(\()p Ft(\025)20 b Fy(+)g Ft(')2035 701 y FF(l)2062 686 y Fy(\))2097 701 y FF(l)q FG(2)p FF(L)2218 686 y Fy(.)p 3371 678 67 67 v 589 848 a(Note)38 b(that)e(w)m(e)g(ha)m (v)m(e)i(a)e(simple)e(description)f(of)j(the)h(range)f(of)g(the)g(em)m (b)s(edding)448 961 y(\(5.2\):)k(this)24 b(is)f(the)i(quotien)m(t)g(of) g(the)g(space)g(of)g(the)g(elemen)m(ts)g(of)g(the)g(form)f(\()p Ft(\025)9 b Fy(+)g Ft(')3254 976 y FF(l)3281 961 y Fy(\))3316 976 y FF(l)q FG(2)p FF(L)448 1074 y Fy(with)29 b Ft(\025)d Fl(2)f Fp(C)54 b Fy(and)30 b(\()p Ft(')1187 1089 y FF(l)1213 1074 y Fy(\))1248 1089 y FF(l)q FG(2)p FF(L)1395 1074 y Fl(2)25 b Ft(C)1546 1088 y Fq(0)1585 1074 y Fy(\()p Ft(X)7 b Fy(\))1737 1041 y Fq([)p FF(L)p Fq(])1830 1074 y Fy(.)448 1265 y Fz(5.2.)91 b Fy(W)-8 b(e)33 b(are)f(no)m(w)g(ready)f (to)i(in)m(tro)s(duce)e(the)h Ft(C)2203 1233 y FG(\003)2242 1265 y Fy(-algebra)g(of)g(energy)g(observ)-5 b(ables)448 1378 y(corresp)s(onding)21 b(to)i(quan)m(tum)f(systems)h(with)e(in)m (teractions)h(ha)m(ving)h(sparse)f(supp)s(orts.)448 1491 y(This)28 b(will)e(b)s(e)j(done)g(b)m(y)g(\\quan)m(tifying")f(the)h (algebra)h(of)f(p)s(oten)m(tials,)g(i.e.)40 b(b)m(y)58 b(taking)448 1604 y(the)37 b(crossed)g(pro)s(duct)f(of)h Ft(C)1455 1618 y FF(L)1507 1604 y Fy(\()p Ft(X)7 b Fy(\))38 b(b)m(y)e(the)h(action)g(of)g(translations)f(on)h(the)g(lo)s(cally)448 1717 y(compact)24 b(group)e Ft(X)7 b Fy(.)39 b(The)22 b(second)h(equalit)m(y)f(b)s(elo)m(w)f(is)h(a)h(consequence)g(of)g(the) g(general)448 1830 y(Theorem)30 b(3.12.)448 2018 y Fz(De\014nition)36 b(5.4)46 b Fs(C)1177 2032 y FF(L)1254 2018 y Fy(:=)25 b Ft(C)1440 2032 y FF(L)1492 2018 y Fy(\()p Ft(X)7 b Fy(\))22 b Fj(o)e Ft(X)32 b Fy(=)25 b([)-12 b([)p Ft(C)2063 2032 y FF(L)2116 2018 y Fy(\()p Ft(X)7 b Fy(\))21 b Fl(\001)g Ft(C)2400 2032 y Fq(0)2439 2018 y Fy(\()p Ft(X)2556 1985 y FG(\003)2597 2018 y Fy(\)])-12 b(])448 2205 y(In)30 b(the)h(same)f(manner)g(w)m(e)h(ma)m(y)g(de\014ne)e(the)i(smaller)e Ft(C)2411 2172 y FG(\003)2450 2205 y Fy(-algebra)h(\(without)g(unit\)) 448 2410 y(\(5.3\))437 b Fs(C)1131 2424 y FF(L;)p Fq(0)1264 2410 y Fy(:=)25 b Ft(C)1450 2424 y FF(L;)p Fq(0)1557 2410 y Fy(\()p Ft(X)7 b Fy(\))21 b Fj(o)f Ft(X)33 b Fy(=)25 b([)-12 b([)p Ft(C)2128 2424 y FF(L;)p Fq(0)2235 2410 y Fy(\()p Ft(X)7 b Fy(\))22 b Fl(\001)e Ft(C)2519 2424 y Fq(0)2559 2410 y Fy(\()p Ft(X)2676 2372 y FG(\003)2716 2410 y Fy(\)])-12 b(])p Ft(:)448 2614 y Fy(Then,)30 b(b)m(y)g(\(3.16\)) j(w)m(e)e(can)f(write)g Fs(C)1694 2628 y FF(L)1776 2614 y Fy(as)h(a)g(linear)e(direct)g(sum)448 2818 y(\(5.4\))892 b Fs(C)1586 2832 y FF(L)1663 2818 y Fy(=)25 b Ft(C)1824 2832 y Fq(0)1864 2818 y Fy(\()p Ft(X)1981 2780 y FG(\003)2021 2818 y Fy(\))20 b(+)g Fs(C)2228 2832 y FF(L;)p Fq(0)2335 2818 y Ft(:)448 3022 y Fy(The)h(algebra)f Fs(C)995 3036 y FF(L;)p Fq(0)1123 3022 y Fy(is)g(an)h(ideal)e(of)i Fs(C)1684 3036 y FF(L)1757 3022 y Fy(and)f Fs(C)1985 3036 y FF(L)2063 3022 y Fl(!)25 b Ft(C)2244 3036 y Fq(0)2283 3022 y Fy(\()p Ft(X)2400 2989 y FG(\003)2441 3022 y Fy(\))c(is)e(a)j (surjectiv)m(e)e(morphism)448 3135 y(whic)m(h)34 b(giv)m(es)h(the)g (pure)f(kinetic)f(energy)i(part,)h(and)f Fs(C)2378 3149 y FF(L)2430 3135 y Ft(=)p Fs(C)2536 3149 y FF(L;)p Fq(0)2676 3110 y Fl(\030)2676 3139 y Fy(=)2779 3135 y Ft(C)2844 3149 y Fq(0)2884 3135 y Fy(\()p Ft(X)3001 3102 y FG(\003)3041 3135 y Fy(\).)54 b(On)34 b(the)448 3248 y(other)45 b(hand,)j Ft(C)1036 3262 y Fq(0)1075 3248 y Fy(\()p Ft(X)7 b Fy(\))46 b(is)e(a)i(stable)e(ideal)g(of)h Ft(C)2162 3262 y FF(L)2214 3248 y Fy(\()p Ft(X)7 b Fy(\))46 b(th)m(us)e(the)h(crossed)g(pro)s (duct)448 3361 y(subalgebra)30 b Ft(C)969 3375 y Fq(0)1009 3361 y Fy(\()p Ft(X)7 b Fy(\))21 b Fj(o)g Ft(X)38 b Fy(is)30 b(also)h(an)f(ideal)g(of)h Fs(C)2171 3375 y FF(L)2223 3361 y Fy(.)43 b(W)-8 b(e)32 b(recall)e(that)h Ft(C)2953 3375 y Fq(0)2993 3361 y Fy(\()p Ft(X)7 b Fy(\))22 b Fj(o)e Ft(X)33 b Fy(=)448 3474 y Fp(K)15 b Fy(\()q Ft(X)7 b Fy(\))37 b(is)29 b(the)i(algebra)f(of)h(compact)g(op)s(erators)g (\(Prop)s(osition)e(3.14\).)589 3637 y(W)-8 b(e)32 b(men)m(tion)e(no)m (w)g(some)h(elemen)m(ts)g(of)f(the)h(m)m(ultiplier)c(algebra)j(of)h Fs(C)3046 3651 y FF(L)3098 3637 y Fy(:)448 3750 y(\(i\))c(if)f Ft(')f Fy(:)h Ft(X)33 b Fl(!)25 b Fp(C)50 b Fy(is)26 b(b)s(ounded,)g(Borel)h(and)f(tends)g(to)i(zero)f(at)h(in\014nit)m(y)-8 b(,)26 b(then)g Ft(')p Fy(\()p Ft(Q)p Fy(\))p Ft(T)448 3862 y Fy(and)k Ft(T)13 b(')p Fy(\()p 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Fy(resp)s(ectiv)m(ely\).)589 4477 y(The)40 b(general)g(theory)h(exp)s (osed)f(in)e(section)j(3)f(allo)m(ws)g(us)f(to)i(giv)m(e)g(a)g (complete)448 4590 y(c)m(haracterization)32 b(b)s(oth)e(of)h(the)g (quotien)m(t)g(algebras)g Fs(C)2364 4604 y FF(L;)p Fq(0)2471 4590 y Ft(=)p Fp(K)15 b Fy(\()p Ft(X)8 b Fy(\))37 b(and)30 b Fs(C)3009 4604 y FF(L)3061 4590 y Ft(=)p Fp(K)16 b Fy(\()p Ft(X)7 b Fy(\))37 b(in)448 4703 y(terms)30 b(of)g(m)m(uc)m(h)g (simpler)e(ob)5 b(jects)31 b(in)m(v)m(olving)d(only)h(the)h(compact)i (op)s(erator)e(algebra)448 4816 y Fp(K)15 b Fy(\()q Ft(X)7 b Fy(\))37 b(and)30 b(the)g(t)m(w)m(o-b)s(o)s(dy)h(algebra)f(\(see)h (\(4.3\)\):)1897 5225 y(46)p eop %%Page: 47 47 47 46 bop 448 573 a Fz(Theorem)34 b(5.5)46 b Fo(The)41 b(quotient)g(algebr)-5 b(a)42 b Fs(C)2001 587 y FF(L;)p Fq(0)2108 573 y Ft(=)p Fp(K)16 b Fy(\()p Ft(X)7 b Fy(\))47 b Fo(is)41 b(c)-5 b(anonic)g(al)5 b(ly)43 b(isomorphic)448 686 y(to)i(the)g Ft(L)p Fo(-asymptotic)h(algebr)-5 b(a)46 b Fp(K)15 b Fy(\()p Ft(X)8 b Fy(\))1834 653 y FG(h)q FF(L)p FG(i)1992 686 y Fo(of)45 b(the)f(algebr)-5 b(a)46 b(of)f(c)-5 b(omp)g(act)47 b(op)-5 b(er)g(ators)448 799 y Fp(K)15 b Fy(\()q Ft(X)7 b Fy(\))q Fo(.)47 b(One)32 b(has)i(a)f(natur)-5 b(al)34 b(emb)-5 b(e)g(dding:)448 1039 y Fy(\(5.5\))1304 1004 y Fs(C)1365 1018 y FF(L)1402 938 y Fm(.)1461 1082 y Fp(K)15 b Fy(\()p Ft(X)7 b Fy(\))1743 1039 y Ft(,)-15 b Fl(!)1901 1004 y Fp(T)p Fy(\()p Ft(X)7 b Fy(\))2112 971 y Fq([)p FF(L)p Fq(])2192 938 y Fm(.)2250 1082 y Fp(K)15 b Fy(\()q Ft(X)7 b Fy(\))2468 1049 y Fq(\()q FF(L)p Fq(\))448 1273 y Fz(Pro)s(of:)48 b Fy(The)36 b(\014rst)g (assertion)h(follo)m(ws)f(from)g(Prop)s(osition)f(3.10)j(b)s(ecause)f (of)g(\(5.1\).)448 1386 y(In)24 b(order)h(to)g(pro)m(v)m(e)h(the)f (second)g(assertion)f(w)m(e)h(start)h(with)d(the)i(im)m(b)s(edding)d (\(5.2\))27 b(and)448 1498 y(use)j(\(3.15\),)j(Prop)s(osition)c(3.7)i (and)f(\(3.18\))j(to)e(get)520 1690 y Fs(C)581 1704 y FF(L)618 1624 y Fm(.)676 1768 y Fp(K)15 b Fy(\()q Ft(X)7 b Fy(\))956 1725 y Fl(\021)1082 1690 y Fy(\()p Ft(C)1182 1704 y FF(L)1235 1690 y Fy(\()p Ft(X)g Fy(\))21 b Fj(o)f Ft(X)7 b Fy(\))1602 1624 y Fm(.)1660 1768 y Fy(\()p Ft(C)1760 1782 y Fq(0)1800 1768 y Fy(\()p Ft(X)g Fy(\))21 b Fj(o)f Ft(X)7 b Fy(\))2238 1725 y(=)2364 1690 y Ft(C)2429 1704 y FF(L)2481 1690 y Fy(\()p Ft(X)g Fy(\))2619 1624 y Fm(.)2677 1768 y Ft(C)2742 1782 y Fq(0)2782 1768 y Fy(\()p Ft(X)g Fy(\))2955 1725 y Fj(o)20 b Ft(X)633 1942 y(,)-15 b Fl(!)847 1907 y Ft(C)912 1921 y FG(1)987 1907 y Fy(\()p Ft(X)7 b Fy(\))1139 1874 y Fq([)p FF(L)p Fq(])1216 1841 y Fm(.)1275 1985 y Ft(C)1340 1999 y Fq(0)1379 1985 y Fy(\()p Ft(X)g Fy(\))1531 1952 y Fq(\()p FF(L)p Fq(\))1690 1942 y Fj(o)20 b Ft(X)63 b Fy(=)2045 1907 y(\()p Ft(C)2145 1921 y FG(1)2220 1907 y Fy(\()p Ft(X)7 b Fy(\))2372 1874 y Fq([)p FF(L)p Fq(])2485 1907 y Fj(o)20 b Ft(X)7 b Fy(\))2678 1841 y Fm(.)2737 1985 y Fy(\()p Ft(C)2837 1999 y Fq(0)2877 1985 y Fy(\()p Ft(X)g Fy(\))3029 1952 y Fq(\()p FF(L)p Fq(\))3157 1985 y Fj(o)20 b Ft(X)7 b Fy(\))648 2159 y(=)847 2124 y(\()p Ft(C)947 2138 y FG(1)1022 2124 y Fy(\()p Ft(X)g Fy(\))22 b Fj(o)e Ft(X)7 b Fy(\))1404 2091 y Fq([)p FF(L)p Fq(])1481 2058 y Fm(.)1539 2202 y Fy(\()p Ft(C)1639 2216 y Fq(0)1679 2202 y Fy(\()p Ft(X)g Fy(\))21 b Fj(o)f Ft(X)7 b Fy(\))2060 2169 y Fq(\()p FF(L)p Fq(\))2224 2159 y Fy(=)2350 2124 y Fp(T)p Fy(\()p Ft(X)g Fy(\))2561 2091 y Fq([)p FF(L)p Fq(])2640 2058 y Fm(.)2699 2202 y Fp(K)15 b Fy(\()p Ft(X)7 b Fy(\))2917 2169 y Fq(\()p FF(L)p Fq(\))3060 2159 y Ft(:)p 3168 2151 67 67 v 589 2388 a Fy(Theorem)22 b(5.5)h(is)d(the)i(main)f(result)f(of)i(this)f(Section:)36 b(it)21 b(allo)m(ws,)i(via)e(Theorem)h(2.2,)448 2501 y(to)30 b(compute)e(the)h(essen)m(tial)f(sp)s(ectrum)g(of)g(a)h (hamiltonian)d(a\016liated)i(to)h(the)g(algebra)448 2614 y(of)d(energy)f(observ)-5 b(ables)24 b Fs(C)1366 2628 y FF(L)1444 2614 y Fy(in)g(terms)h(of)g(sp)s(ectra)g(of)h(hamiltonians) c(a\016liated)j(to)h(the)448 2727 y(t)m(w)m(o-b)s(o)s(dy)31 b(algebra.)589 2840 y(Notice)i(that,)g(as)f(in)f(the)h(ab)s(elian)e (case,)j(w)m(e)g(ha)m(v)m(e)g(a)f(precise)f(description)f(of)i(the)448 2952 y(range)g(of)f(the)g(em)m(b)s(edding)e(\(5.5\):)43 b(it)31 b(is)f(the)h(quotien)m(t)g(with)e(resp)s(ect)i(to)h Fp(K)15 b Fy(\()p Ft(X)8 b Fy(\))3220 2920 y Fq(\()p FF(L)p Fq(\))3364 2952 y Fy(of)448 3065 y(the)30 b(subspace)f(of)h Fp(T)p Fy(\()p Ft(X)7 b Fy(\))1297 3032 y Fq([)p FF(L)p Fq(])1422 3065 y Fy(consisting)28 b(of)i(sequences)g(of)g(the)g(form)f (\()p Ft( )s Fy(\()p Ft(P)13 b Fy(\))21 b(+)e Ft(K)3255 3080 y FF(l)3281 3065 y Fy(\))3316 3080 y FF(l)q FG(2)p FF(L)448 3178 y Fy(for)30 b(some)h Ft( )e Fl(2)c Ft(C)1054 3192 y Fq(0)1093 3178 y Fy(\()p Ft(X)1210 3145 y FG(\003)1250 3178 y Fy(\))31 b(and)f(\()p Ft(K)1605 3193 y FF(l)1631 3178 y Fy(\))1666 3193 y FF(l)q FG(2)p FF(L)1813 3178 y Fl(2)25 b Fp(K)15 b Fy(\()p Ft(X)8 b Fy(\))2117 3145 y Fq([)p FF(L)p Fq(])2215 3178 y Fy(.)589 3301 y(If)43 b Ft(H)49 b Fy(is)42 b(an)g(observ)-5 b(able)42 b(a\016liated)g(to)i Fs(C)2086 3315 y FF(L)2181 3301 y Fy(and)e(if)2469 3278 y Fm(c)2465 3301 y Ft(H)65 b Fy(is)42 b(its)g(image)h(through)448 3413 y(the)33 b(canonical)g(morphism)d Fs(C)1497 3427 y FF(L)1579 3413 y Fl(!)f Fs(C)1760 3427 y FF(L)1812 3413 y Ft(=)p Fp(K)16 b Fy(\()p Ft(X)8 b Fy(\),)40 b(then)32 b(there)h(is)f(a)h(family)f(\()p Ft(H)3149 3428 y FF(l)3175 3413 y Fy(\))3210 3428 y FF(l)q FG(2)p FF(L)3364 3413 y Fy(of)448 3526 y(observ)-5 b(ables)32 b(a\016liated)f(to)i(the)f(t)m (w)m(o-b)s(o)s(dy)h(algebra)f Fp(T)p Fy(\()p Ft(X)7 b Fy(\))35 b(suc)m(h)d(that)g(the)h(quotien)m(t)448 3649 y(of)557 3580 y Fm(Q)643 3675 y FF(l)q FG(2)p FF(L)779 3649 y Ft(H)855 3664 y FF(l)916 3649 y Fy(with)h(resp)s(ect)h(to)i(the) e(ideal)g Fp(K)15 b Fy(\()p Ft(X)7 b Fy(\))2163 3616 y Fq(\()p FF(L)p Fq(\))2311 3649 y Fy(b)s(e)35 b(equal)g(to)h(the)g (image)f(of)3343 3626 y Fm(c)3339 3649 y Ft(H)448 3761 y Fy(through)c(the)i(em)m(b)s(edding)d(\(5.5\).)47 b(Suc)m(h)31 b(a)h(family)e(\()p Ft(H)2358 3776 y FF(l)2384 3761 y Fy(\))2419 3776 y FF(l)q FG(2)p FF(L)2572 3761 y Fy(will)g(b)s(e)h (called)g(a)h Fo(r)-5 b(epr)g(e-)448 3874 y(sentative)36 b Fy(of)29 b Ft(H)7 b Fy(.)40 b(By)29 b(the)f(discussion)e(ab)s(o)m(v)m (e)k(w)m(e)f(ha)m(v)m(e)2397 3806 y Fm(Q)2483 3901 y FF(l)q FG(2)p FF(L)2604 3874 y Fy(\()p Ft(H)2715 3889 y FF(l)2757 3874 y Fl(\000)17 b Ft(z)t Fy(\))2926 3841 y FG(\000)p Fq(1)3046 3874 y Fl(2)25 b Fp(T)p Fy(\()p Ft(X)7 b Fy(\))3343 3841 y Fq([)p FF(L)p Fq(])448 3987 y Fy(and)40 b(the)g(comp)s(onen)m(t)g(of)g(\()p Ft(H)1499 4002 y FF(l)1551 3987 y Fl(\000)26 b Ft(z)t Fy(\))1729 3954 y FG(\000)p Fq(1)1864 3987 y Fy(in)39 b Ft(C)2045 4001 y Fq(0)2084 3987 y Fy(\()p Ft(X)2201 3954 y FG(\003)2241 3987 y Fy(\))h(is)f(indep)s(enden)m(t)f(of)i Ft(l)j Fl(2)e Ft(L)p Fy(,)h(so)448 4100 y Fr(\033)502 4114 y Fq(ess)593 4100 y Fy(\()p Ft(H)704 4115 y FF(l)730 4100 y Fy(\))36 b(is)f(indep)s(enden)m(t)f(of)i Ft(l)r Fy(.)57 b(Th)m(us)35 b(the)h(next)g(result)e(is)h(a)h(consequence)h(of)f(the)448 4213 y(Theorem)30 b(2.2.)448 4401 y Fz(Theorem)k(5.6)46 b Fo(If)34 b Ft(H)41 b Fo(is)34 b(an)g(observable)g(a\016liate)-5 b(d)36 b(to)f Fs(C)2493 4415 y FF(L)2579 4401 y Fo(and)f Fl(f)p Ft(H)2877 4416 y FF(l)2903 4401 y Fl(g)2948 4416 y FF(l)q FG(2)p FF(L)3104 4401 y Fo(is)f(a)i(r)-5 b(ep-)448 4514 y(r)g(esentative)34 b(of)f Ft(H)7 b Fo(,)32 b(then)1394 4745 y Fr(\033)1448 4759 y Fq(ess)1540 4745 y Fy(\()p Ft(H)7 b Fy(\))25 b(=)1856 4658 y Fm(\\)1838 4852 y Fv(F)8 b Fx(\032)p Fv(L)1814 4900 y(F)g Fk(\014nite)p 2014 4638 453 4 v 2066 4658 a Fm([)2014 4860 y FF(l)q FG(2)p FF(L)p FG(n)p FF(F)2235 4745 y Fr(\033)m Fy(\()p Ft(H)2405 4760 y FF(l)2430 4745 y Fy(\))q Ft(:)1897 5225 y Fy(47)p eop %%Page: 48 48 48 47 bop 589 573 a Fy(Let)32 b(us)f(giv)m(e)h(no)m(w)g(a)g(second)f (description)f(of)h(the)h(quotien)m(t)g(algebra)f Fs(C)3090 587 y FF(L)3142 573 y Ft(=)p Fp(K)16 b Fy(\()p Ft(X)8 b Fy(\),)448 686 y(based)32 b(on)h(the)f(formalism)f(w)m(e)i(dev)m (elopp)s(ed)e(in)g Fl(x)p Fy(2.5.)49 b(According)32 b(to)h(the)g (notations)448 799 y(in)m(tro)s(duced)25 b(there,)j(w)m(e)f(shall)e (denote)i(b)m(y)1918 776 y Fm(e)1905 799 y Ft(L)41 b Fy(the)27 b(set)g(of)g(ultra\014lters)e(on)h Ft(L)h Fy(\014ner)e(than) 448 912 y(the)31 b(F)-8 b(r)m(\023)-43 b(ec)m(het)33 b(\014lter;)1183 889 y Fm(e)1170 912 y Ft(L)45 b Fy(is)30 b(a)g(compact)i(top)s(ological)e(space.)448 1099 y Fz(Theorem)k(5.7)46 b Fo(F)-7 b(or)36 b(e)-5 b(ach)35 b Ft(S)f Fl(2)29 b Fs(C)1690 1113 y FF(L)1776 1099 y Fo(and)36 b Fj({)c Fl(2)2151 1076 y Fm(e)2137 1099 y Ft(L)50 b Fo(the)35 b(limit)g Fy(s)15 b(-lim)2828 1114 y FF(l)q(;)p Fi({)2938 1099 y Ft(U)3000 1114 y FF(l)3026 1099 y Ft(S)5 b(U)3159 1066 y FG(\003)3149 1127 y FF(l)3228 1099 y Fy(=)28 b Ft(S)3383 1113 y Fi({)448 1212 y Fo(exists)e(in)g(the)g(str)-5 b(ong)26 b(op)-5 b(er)g(ator)29 b(top)-5 b(olo)g(gy)29 b(and)d(b)-5 b(elongs)26 b(to)g Fp(T)p Fy(\()p Ft(X)7 b Fy(\))29 b(=)c Ft(C)2873 1226 y Fq(0)2912 1212 y Fy(\()p Ft(X)3029 1179 y FG(\003)3069 1212 y Fy(\))5 b(+)g Fp(K)15 b Fy(\()p Ft(X)8 b Fy(\))p Fo(.)448 1325 y(The)33 b(c)-5 b(omp)g(onent)36 b(of)d Ft(S)1251 1339 y Fi({)1338 1325 y Fo(in)g Ft(C)1515 1339 y Fq(0)1554 1325 y Fy(\()p Ft(X)1671 1292 y FG(\003)1711 1325 y Fy(\))g Fo(is)g(e)-5 b(qual)34 b(to)f(that)h(of)f Ft(S)38 b Fo(in)33 b Ft(C)2781 1339 y Fq(0)2820 1325 y Fy(\()p Ft(X)2937 1292 y FG(\003)2978 1325 y Fy(\))p Fo(.)43 b(The)33 b(map)448 1438 y Ft(S)50 b Fl(7!)44 b Fy(\()p Ft(S)780 1452 y Fi({)834 1438 y Fy(\))869 1471 y Fi({)s FG(2)976 1455 y Fh(e)966 1471 y FF(L)1074 1438 y Fo(is)f(a)h(morphism)h(of)f Fs(C)1885 1452 y FF(L)1980 1438 y Fo(into)g Ft(C)7 b Fy(\()2299 1415 y Fm(e)2286 1438 y Ft(L)15 b Fy(;)g Fp(T)p Fy(\()p Ft(X)7 b Fy(\)\))47 b Fo(with)d(kernel)f Fp(K)15 b Fy(\()p Ft(X)8 b Fy(\))p Fo(.)448 1551 y(Its)38 b(r)-5 b(ange)38 b(is)f(e)-5 b(qual)38 b(to)g(the)g(set)f(of)h Fy(\()p Ft(S)1796 1565 y Fi({)1850 1551 y Fy(\))1885 1584 y Fi({)s FG(2)1992 1568 y Fh(e)1982 1584 y FF(L)2084 1551 y Fo(such)f(that)i (the)f(c)-5 b(omp)g(onent)39 b(of)f Ft(S)3267 1565 y Fi({)3358 1551 y Fo(in)448 1664 y Ft(C)513 1678 y Fq(0)553 1664 y Fy(\()p Ft(X)670 1631 y FG(\003)710 1664 y Fy(\))33 b Fo(is)f(indep)-5 b(endent)35 b(of)d Fj({)t Fo(.)448 1851 y Fz(Pro)s(of:)48 b Fy(One)32 b(has)h(a)g(unique)e(decomp)s (osition)h(of)h Ft(S)k Fy(in)m(to)c(a)h(sum)d Ft(T)k Fy(+)22 b Ft(S)3016 1818 y FG(0)3072 1851 y Fy(with)31 b Ft(T)43 b Fl(2)448 1964 y Ft(C)513 1978 y Fq(0)553 1964 y Fy(\()p Ft(X)670 1931 y FG(\003)710 1964 y Fy(\))c(and)g Ft(S)1031 1931 y FG(0)1094 1964 y Fl(2)h Fs(C)1256 1978 y FF(L;)p Fq(0)1363 1964 y Fy(.)67 b(Since)38 b Ft(U)1763 1979 y FF(l)1789 1964 y Ft(T)13 b(U)1927 1931 y FG(\003)1917 1992 y FF(l)2006 1964 y Fy(=)40 b Ft(T)13 b Fy(,)41 b(it)e(su\016ces)g (to)h(consider)e Ft(T)53 b Fy(=)39 b(0.)448 2077 y(Then)28 b(b)m(y)g(\(iii\))f(of)i(Lemma)g(5.1)g(and)f(\(5.3\))j(it)d(su\016ces)g (to)h(tak)m(e)h Ft(S)h Fy(=)25 b 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Ft(m)p Fy(\))40 b Fl(6)p Fy(=)f(0)h(then)e Ft(l)k Fl(2)d Ft(m)26 b Fy(+)g(\(\003)g Fl(\000)g Ft(K)7 b Fy(\))39 b(and)448 2980 y(\003)18 b Fl(\000)g Ft(K)36 b Fy(is)28 b(a)h(compact)i(set.)41 b(So)29 b(if)f Ft(l)j Fy(is)d(large)h(enough)g(then)g Ft(')2588 2994 y Fq(0)2628 2980 y Fy(\()p Ft(x)18 b Fy(+)g Ft(l)h Fl(\000)f Ft(m)p Fy(\))25 b Fl(6)p Fy(=)g(0)30 b(only)448 3093 y(if)h Ft(l)f Fy(=)e Ft(m)k Fy(\()p Ft(L)g Fy(b)s(eing)f(sparse\).)46 b(So)32 b(for)g(large)g Ft(l)i Fy(one)e(has)g(either)g Ft(\034)2713 3108 y FG(\000)p FF(l)2793 3093 y Ft(')p Fy(\()p Ft(x)p Fy(\))e(=)d Ft(')3161 3107 y Fq(0)3201 3093 y Fy(\()p Ft(x)p Fy(\))33 b(or)448 3206 y Ft(\034)488 3221 y FG(\000)p FF(l)569 3206 y Ft(')p Fy(\()p Ft(x)p Fy(\))26 b(=)f(0)31 b(\(indep)s(enden)m(tly)d(of)i Ft(x)c Fl(2)e Ft(K)7 b Fy(\).)589 3319 y(W)-8 b(e)32 b(ha)m(v)m(e)g(th)m(us)f(sho)m(wn)f(that)h(the)g(limit)e(s)15 b(-lim)2212 3334 y FF(l)q(;)p Fi({)2323 3319 y Ft(U)2385 3334 y FF(l)2411 3319 y Ft(S)5 b(U)2544 3286 y FG(\003)2534 3347 y FF(l)2609 3319 y Fy(:=)26 b Ft(S)2787 3333 y Fi({)2872 3319 y Fy(exists)k(for)h(eac)m(h)448 3444 y Fj({)38 b Fl(2)656 3421 y Fm(e)643 3444 y Ft(L)15 b Fy(.)58 b(The)35 b(argumen)m(t)i(also)f(giv)m(es)g(the)g(explicit)f(form)h(of)g(the)g (limit)e(for)i(a)g(class)448 3557 y(of)45 b(op)s(erators)g Ft(S)50 b Fy(whic)m(h)43 b(is)h(dense)g(in)f Fs(C)1912 3571 y FF(L)1965 3557 y Fy(.)83 b(Namely)-8 b(,)49 b(assume)44 b(that)i Ft(S)j Fy(is)44 b(of)h(the)448 3670 y(form)37 b Ft( )729 3684 y Fq(0)769 3670 y Fy(\()p Ft(P)13 b Fy(\))26 b(+)1031 3602 y Fm(P)1127 3628 y FF(n)1127 3697 y(i)p Fq(=1)1261 3670 y Ft(')1320 3684 y FF(i)1348 3670 y Fy(\()p Ft(Q)p Fy(\))p Ft( )1549 3684 y FF(i)1578 3670 y Fy(\()p Ft(P)13 b Fy(\))38 b Fl(\021)f Ft(T)h Fy(+)24 b Ft(S)2112 3637 y FG(0)2136 3670 y Fy(,)39 b(with)d Ft( )2473 3684 y Fq(1)2513 3670 y Ft(;)15 b(:)g(:)g(:)i(;)e( )2774 3684 y FF(n)2858 3670 y Fl(2)37 b Ft(C)3021 3684 y Fq(0)3061 3670 y Fy(\()p Ft(X)3178 3637 y FG(\003)3218 3670 y 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4329 y Fq(0)1610 4315 y Fy(\()p Ft(P)13 b Fy(\))21 b(+)1907 4201 y FF(n)1863 4229 y Fm(X)1871 4424 y FF(i)p Fq(=1)2009 4315 y Ft(')2068 4334 y FF(ij)t Fq(\()p FF(i)p Fq(\))2208 4315 y Fy(\()p Ft(Q)p Fy(\))p Ft( )2409 4329 y FF(i)2438 4315 y Fy(\()p Ft(P)13 b Fy(\))p Ft(:)448 4601 y Fy(Th)m(us)31 b Ft(S)736 4615 y Fi({)817 4601 y Fl(2)26 b Fp(T)p Fy(\()p Ft(X)7 b Fy(\))35 b(and)c(its)f(pro)5 b(jection)31 b(on)h Ft(C)2081 4615 y Fq(0)2120 4601 y Fy(\()p Ft(X)2237 4568 y FG(\003)2277 4601 y Fy(\))g(is)f Ft( )2496 4615 y Fq(0)2535 4601 y Fy(\()p Ft(P)13 b Fy(\),)33 b(whic)m(h)d(is)h(the)g(com-)448 4714 y(p)s(onen)m(t)43 b(of)h Ft(S)49 b Fy(in)42 b Ft(C)1170 4728 y Fq(0)1209 4714 y Fy(\()p Ft(X)1326 4681 y FG(\003)1366 4714 y Fy(\).)81 b(This)42 b(remains)g(v)-5 b(alid)42 b(for)h(all)f Ft(S)49 b Fy(b)m(y)43 b(con)m(tin)m(uit)m(y)h(and)448 4827 y(densit)m(y)-8 b(.)1897 5225 y(48)p eop %%Page: 49 49 49 48 bop 589 574 a Fy(Finally)-8 b(,)36 b(consider)f(the)h(image)1718 550 y Fm(c)1713 574 y Ft(S)1774 547 y FG(0)1849 574 y Fy(of)g Ft(S)2019 541 y FG(0)2078 574 y Fy(in)f Fp(K)15 b Fy(\()p Ft(X)7 b Fy(\))2408 541 y FG(h)p FF(L)p FG(i)2557 574 y Fy(giv)m(en)36 b(b)m(y)f(Theorem)h(5.5)448 697 y(and)c(iden)m(tify)f Fp(K)15 b Fy(\()p Ft(X)8 b Fy(\))1178 664 y FG(h)q FF(L)p FG(i)1320 672 y Fl(\030)1320 701 y Fy(=)1419 697 y Ft(C)f Fy(\()1539 674 y Fm(e)1526 697 y Ft(L)15 b Fy(;)g Fp(K)h Fy(\()p Ft(X)7 b Fy(\))q(\),)39 b(cf.)47 b(\(2.9\).)i(Then)2604 673 y Fm(c)2600 697 y Ft(S)2661 671 y FG(0)2731 697 y Fy(will)30 b(b)s(e)i(the)h(family)448 810 y(of)e(op)s(erators)f Ft(S)1010 824 y Fi({)1095 810 y Fy(de\014ned)f(b)m(y)h(\(5.6\),)i(so)f(the)g(theorem)f(is)g(pro)m(v)m (ed.)p 3371 802 67 67 v 589 971 a(The)g(follo)m(wing)f(result)g(is)h(a) h(straigh)m(tforw)m(ard)f(consequence)h(of)f(Theorem)h(5.7.)448 1153 y Fz(Theorem)j(5.8)46 b Fo(L)-5 b(et)49 b Ft(H)54 b Fo(b)-5 b(e)48 b(an)h(observable)f(a\016liate)-5 b(d)50 b(to)f Fs(C)2667 1167 y FF(L)2719 1153 y Fo(.)88 b(Then)48 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4706 y Fy(.)40 b(As)26 b(usual,)g(b)m(y)2721 4688 y Ft(\037)2778 4720 y FF(L)2826 4697 y Fv(c)2826 4743 y Fk(\003)2876 4706 y Fy(\()p Ft(Q)p Fy(\))h(w)m(e)f(denote)448 4819 y(the)31 b(op)s(erator)g(of)f(m)m(ultiplication)d(b)m(y)k(the)f(c)m (haracteristic)h(function)e(of)i(the)g(set)f Ft(L)3320 4786 y FF(c)3320 4846 y Fq(\003)3373 4819 y Fy(.)1897 5225 y(51)p eop %%Page: 52 52 52 51 bop 448 573 a Fz(Theorem)34 b(5.11)47 b Fo(A)32 b(b)-5 b(ounde)g(d)34 b(op)-5 b(er)g(ator)36 b Ft(T)45 b Fo(on)33 b Ft(L)2218 540 y Fq(2)2257 573 y Fy(\()p Ft(X)7 b Fy(\))34 b Fo(b)-5 b(elongs)33 b(to)g Fs(C)2924 587 y FF(L;)p Fq(0)3064 573 y Fo(i\013)448 686 y Fy(\(i\))g(lim)703 700 y FF(x)p FG(!)p Fq(0)868 686 y Fl(k)p Fy(\()p Ft(U)1010 700 y FF(x)1075 686 y Fl(\000)20 b Fy(1\))p Ft(T)1312 653 y Fq(\()p FG(\003)p Fq(\))1407 686 y Fl(k)25 b Fy(=)g(0)p Fo(,)448 799 y Fy(\(ii\))32 b(lim)728 814 y FF(k)r FG(!)p Fq(0)892 799 y Fl(k)p Ft(V)990 814 y FF(k)1033 799 y Ft(T)13 b(V)1172 766 y FG(\003)1152 826 y FF(k)1232 799 y Fl(\000)20 b Ft(T)13 b Fl(k)25 b Fy(=)g(0)p Fo(,)448 912 y Fy(\(iii\))d Fo(F)-7 b(or)24 b(e)-5 b(ach)25 b Ft(")g(>)g Fy(0)f Fo(ther)-5 b(e)24 b(is)f(a)h(c)-5 b(omp)g(act)26 b(set)d Fy(\003)i Fl(\032)g Ft(X)31 b Fo(such)23 b(that)i Fl(k)2764 894 y Ft(\037)2821 926 y FF(L)2869 903 y Fv(c)2869 949 y Fk(\003)2919 912 y Fy(\()p Ft(Q)p Fy(\))p Ft(T)3127 879 y Fq(\()p FG(\003)p Fq(\))3222 912 y Fl(k)g Ft(<)g(":)448 1130 y Fz(Corollary)35 b(5.12)47 b Fo(A)30 b(b)-5 b(ounde)g(d)32 b(op)-5 b(er)g(ator)33 b Ft(T)44 b Fo(on)31 b Ft(L)2229 1097 y Fq(2)2268 1130 y Fy(\()p Ft(X)7 b Fy(\))32 b Fo(b)-5 b(elongs)31 b(to)g Fs(C)2929 1144 y FF(L)3011 1130 y Fo(if)g(and)g(only)448 1243 y(if)49 b(it)f(satis\014es)i(c)-5 b(onditions)50 b Fy(\(i\),)i(\(ii\))c Fo(of)h(The)-5 b(or)g(em)50 b(5.11)g(and)f(ther)-5 b(e)50 b(exists)3257 1220 y Fm(e)3241 1243 y Ft(T)83 b Fl(2)448 1356 y Ft(C)513 1370 y Fq(0)553 1356 y Fy(\()p Ft(X)670 1323 y FG(\003)710 1356 y Fy(\))49 b Fo(such)g(that)i(for)e(e)-5 b(ach)50 b Ft(")55 b(>)g Fy(0)49 b Fo(ther)-5 b(e)50 b(is)f(a)h(c)-5 b(omp)g(act)51 b(set)e Fy(\003)55 b Fl(\032)g Ft(X)h Fo(with)448 1469 y Fl(k)493 1451 y Ft(\037)551 1483 y FF(L)599 1460 y Fv(c)599 1506 y Fk(\003)648 1469 y Fy(\()p Ft(Q)p Fy(\))15 b(\()p Ft(T)34 b Fl(\000)1034 1446 y Fm(e)1018 1469 y Ft(T)28 b Fy(\))1134 1436 y Fq(\()p FG(\003)p Fq(\))1229 1469 y Fl(k)e Ft(<)f(":)589 1679 y Fy(Notice)45 b(that)e(an)h(equiv)-5 b(alen)m(t)42 b(form)m(ulation)g (of)i(\(ii\))e(is)g(lim)2695 1694 y FF(k)r FG(!)p Fq(0)2859 1679 y Fl(k)p Fy([)p Ft(V)2982 1694 y FF(k)3026 1679 y Ft(;)15 b(T)e Fy(])p Fl(k)47 b Fy(=)f(0.)448 1791 y(W)-8 b(e)33 b(denote)e(b)m(y)g Fs(A)g Fy(the)h(set)f(of)g(op)s(erators)h(v)m (erifying)d(the)i(conditions)f(\(i\)-\(iii\))g(of)h(the)448 1904 y(theorem.)68 b(Clearly)38 b Fs(A)i Fy(is)e(a)i Ft(C)1557 1871 y FG(\003)1596 1904 y Fy(-algebra.)68 b(If)39 b(one)g(replaces)g Ft(U)2697 1918 y FF(x)2780 1904 y Fy(b)m(y)h Ft(V)2969 1919 y FF(k)3051 1904 y Fy(in)e(\(i\))h 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y FF(x)644 2558 y Fl(\000)9 b Fy(1\))p Ft(')p Fy(\()p Ft(Q)p Fy(\))p Ft( )s Fy(\()p Ft(P)k Fy(\))p Fl(k)28 b(\024)d(k)p Ft(')p Fy(\()p Ft(Q)9 b Fy(+)g Ft(x)p Fy(\))g Fl(\000)g Ft(')p Fy(\()p Ft(Q)p Fy(\))p Fl(k)15 b(k)p Ft( )s Fy(\()p Ft(P)e Fy(\))p Fl(k)c Fy(+)g Fl(k)p Ft(')p Fy(\()p Ft(Q)p Fy(\))p Fl(k)15 b(k)p Fy(\()p Ft(U)2944 2572 y FF(x)3002 2558 y Fl(\000)9 b Fy(1\))p Ft( )s Fy(\()p Ft(P)k Fy(\))p Fl(k)p Ft(:)448 2759 y Fy(The)31 b(function)e Ft(')i Fy(is)f(uniformly)d(con)m(tin)m(uous)k(so)g(the)g(\014rst)f (term)g(in)g(the)g(r.h.s.)h(ab)s(o)m(v)m(e)448 2872 y(tends)f(to)h (zero)h(as)e Ft(x)25 b Fl(!)h Fy(0.)41 b(Since)1154 3074 y Fl(k)p Fy(\()p Ft(U)1296 3088 y FF(x)1360 3074 y Fl(\000)20 b Fy(1\))p Ft( )s Fy(\()p Ft(P)13 b Fy(\))p Fl(k)28 b Fy(=)49 b(sup)1903 3153 y FF(k)r FG(2)p FF(X)2052 3134 y Fx(\003)2103 3074 y Fl(j)p Ft(k)s Fy(\()p Ft(x)p Fy(\))21 b Fl(\000)f Fy(1)p Fl(j)15 b(j)p Ft( )s Fy(\()p Ft(k)s Fy(\))p Fl(j)448 3331 y Fy(and)31 b Ft( )s Fy(\()p Ft(k)s Fy(\))e 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Ft(')2091 4313 y Fq(1)2131 4299 y Ft(;)15 b(:)g(:)g(:)h(;)f(')2391 4313 y FF(n)2468 4299 y Fl(2)28 b Ft(C)2622 4313 y FF(L;)p Fq(c)2725 4299 y Fy(\()p Ft(X)7 b Fy(\)\()p Ft(X)g Fy(\))34 b(suc)m(h)e(that)448 4412 y Fl(k)p Ft(T)39 b Fl(\000)681 4344 y Fm(P)777 4370 y FF(n)777 4439 y Fq(1)839 4412 y Ft( )898 4426 y FF(i)926 4412 y Fy(\()p Ft(P)13 b Fy(\))p Ft(')1126 4426 y FF(i)1155 4412 y Fy(\()p Ft(Q)p Fy(\))p Fl(k)39 b Ft(<)f(")p Fy(.)63 b(F)-8 b(or)39 b(\003)f(large)g(enough)g (w)m(e)g(ha)m(v)m(e)h Ft(')2860 4426 y FF(i)2889 4412 y Fy(\()p Ft(Q)p Fy(\))3031 4394 y Ft(\037)3088 4426 y FF(L)3136 4403 y Fv(c)3136 4449 y Fk(\003)3186 4412 y Fy(\()p Ft(Q)p Fy(\))f(=)448 4525 y(0)k(for)f(all)f Ft(i)p Fy(,)k(hence)d Fl(k)p Ft(T)1296 4507 y(\037)1354 4539 y FF(L)1402 4516 y Fv(c)1402 4562 y Fk(\003)1451 4525 y Fy(\()p Ft(Q)p Fy(\))p Fl(k)k Ft(<)d(")p Fy(.)74 b(This)39 b(\014nishes)g(the)i(pro)s(of)g(of)g Fs(C)3076 4539 y FF(L;)p Fq(0)3226 4525 y Fl(\032)i Fs(A)p Fy(.)448 4638 y(The)d(recipro)s(cal)f(assertion)h(is)f(less)h(elemen)m(tary)h (and)e(w)m(e)i(dev)m(ote)h(the)e(rest)g(of)h(the)448 4751 y(subsection)30 b(to)h(its)f(pro)s(of.)589 4863 y(In)g(what)f(follo)m(ws)g(w)m(e)h(shall)e(need)i(three)g(groups)f(of)h (automorphisms)e(of)i Ft(B)5 b Fy(\()p Fn(H)27 b Fy(\),)448 4976 y(namely)i Fl(fU)865 4990 y FF(x)909 4976 y Fl(g)954 4990 y FF(x)p FG(2)p FF(X)1109 4976 y Fy(,)h Fl(fV)1265 4991 y FF(k)1308 4976 y Fl(g)1353 4991 y FF(k)r FG(2)p FF(X)1502 4972 y Fx(\003)1572 4976 y Fy(and)f Fl(fW)1883 4991 y FF(\030)1921 4976 y Fl(g)1966 4991 y FF(\030)s FG(2)p FF(X)5 b FG(\002)p FF(X)2228 4972 y Fx(\003)2268 4976 y Fy(,)30 b(de\014ned)f(on)g(ev)m(ery)h Ft(T)38 b Fl(2)25 b Ft(B)5 b Fy(\()p Fn(H)27 b Fy(\))1897 5225 y(52)p eop %%Page: 53 53 53 52 bop 448 573 a Fy(b)m(y)45 b Fl(U)646 587 y FF(x)689 573 y Fy([)p Ft(T)13 b Fy(])49 b(:=)f Ft(U)1060 587 y FF(x)1104 573 y Ft(T)13 b(U)1242 540 y FG(\003)1232 595 y FF(x)1282 573 y Fy(,)48 b Fl(V)1411 588 y FF(k)1453 573 y Fy([)p Ft(T)13 b Fy(])49 b(:=)f Ft(V)1815 588 y FF(k)1858 573 y Ft(T)13 b(V)1997 540 y FG(\003)1977 601 y FF(k)2080 573 y Fy(and)44 b Fl(W)2361 591 y Fq(\()p FF(x;k)r Fq(\))2566 573 y Fy(:=)49 b Fl(U)2768 587 y FF(x)2811 573 y Fl(V)2867 588 y FF(k)2954 573 y Fy(resp)s(ectiv)m(ely) -8 b(.)448 686 y(Notice)38 b(that)f([)p Fl(U)1025 700 y FF(x)1069 686 y Ft(;)15 b Fl(V)1165 701 y FF(k)1208 686 y Fy(])35 b(=)h(0)g(for)h(eac)m(h)g(couple)f(\()p Ft(x;)15 b(k)s Fy(\))37 b Fl(2)e Ft(X)d Fl(\002)24 b Ft(X)2730 653 y FG(\003)2770 686 y Fy(.)59 b(Hence)37 b Fl(W)3220 701 y FF(\030)3294 686 y Fy(is)f(a)448 799 y(represen)m(tation)23 b(on)f Ft(B)5 b Fy(\()p Fn(H)27 b Fy(\))22 b(of)h(the)g(lo)s(cally)e(compact)i(group)f(\004)j(:=)g Ft(X)12 b Fl(\002)t Ft(X)3016 766 y FG(\003)3079 799 y Fy(equipp)s(ed)448 912 y(with)35 b(the)i(Haar)g(measure)51 b(d)p Ft(\030)39 b Fy(=)50 b(d)o Ft(x)24 b Fl(\012)39 b Fy(d)p Ft(k)s Fy(.)58 b(This)35 b(represen)m(tation)h(is)f(con)m(tin) m(uous)448 1024 y(if)40 b(w)m(e)h(equip)f Ft(B)5 b Fy(\()p Fn(H)26 b Fy(\))41 b(with)f(the)h(strong)g(op)s(erator)g(top)s(ology)g (but)f(it)h(is)f(not)h(norm)448 1137 y(con)m(tin)m(uous.)g(It)30 b(is)g(clear)g(that)448 1327 y(\(5.10\))52 b Ft(C)802 1290 y Fq(0)795 1350 y(u)841 1327 y Fy(\(\004\))10 b(:=)g Fl(f)p Ft(T)39 b Fl(2)25 b Ft(B)5 b Fy(\()p Fn(H)27 b Fy(\))10 b Fl(j)g Fy(\004)25 b Fl(3)g Ft(\030)k Fl(7!)10 b(W)2048 1342 y FF(\030)2086 1327 y Fy([)p Ft(T)j Fy(])25 b Fl(2)10 b Ft(B)5 b Fy(\()p Fn(H)27 b Fy(\))j(is)g(norm)f(con)m(tin)m (uous)p Fl(g)448 1517 y Fy(is)h(a)g Ft(C)687 1484 y FG(\003)726 1517 y Fy(-subalgebra)g(of)h Ft(B)5 b Fy(\()p Fn(H)26 b Fy(\).)589 1630 y(If)k Ft(T)38 b Fl(2)25 b Ft(B)5 b Fy(\()p Fn(H)27 b Fy(\))j(and)g Ft(u)25 b Fl(2)g Ft(L)1549 1597 y Fq(1)1589 1630 y Fy(\()p Ft(X)7 b Fy(\),)31 b Ft(v)e Fl(2)c Ft(L)2018 1597 y Fq(1)2057 1630 y Fy(\()p Ft(X)2174 1597 y FG(\003)2214 1630 y Fy(\),)31 b(w)m(e)g(denote)448 1861 y(\(5.11\))653 b Ft(T)1384 1875 y FF(u;v)1511 1861 y Fy(:=)1632 1738 y Fm(Z)1683 1944 y Fq(\004)1749 1861 y Fl(W)1839 1876 y FF(\030)1877 1861 y Fy([)p Ft(T)13 b Fy(])i(\()p Ft(u)21 b Fl(\012)f Ft(v)s Fy(\)\()p Ft(\030)t Fy(\))31 b(d)p Ft(\030)t(:)448 2092 y Fy(In)f(the)g(next)g(three)g (lemmas)g(w)m(e)g(giv)m(e)h(prop)s(erties)e(of)h(this)f(ob)5 b(ject)31 b(whic)m(h)d(sho)m(w)i(that)448 2205 y Ft(T)501 2219 y FF(u;v)632 2205 y Fy(is)e(a)h(\\regularization")g(of)g Ft(T)41 b Fy(in)28 b(a)h(similar)e(manner)h(in)f(whic)m(h)h(the)h(con)m (v)m(olution)448 2318 y(of)i(a)g(function)e(is)h(a)g(smo)s(othing)g(of) h(this)e(function.)40 b(W)-8 b(e)32 b(men)m(tion)e(that)h(the)g (regular-)448 2431 y(ization)e(in)e Ft(x)p Fy(,)j(realized)e(b)m(y)h Ft(u)p Fy(,)g(is)f(not)h(needed)g(for)g(the)g(pro)s(of)f(of)h(Theorem)g (5.11,)i(but)448 2544 y(will)d(b)s(e)i(useful)e(later)j(on.)448 2719 y Fz(Lemma)i(5.13)46 b Fo(F)-7 b(or)31 b(e)-5 b(ach)31 b Ft(T)38 b Fl(2)25 b Ft(B)5 b Fy(\()p Fn(H)26 b Fy(\))31 b Fo(the)f(fol)5 b(lowing)31 b(statements)g(ar)-5 b(e)31 b(e)-5 b(quivalent:)448 2832 y Fy(\(i\))52 b Fo(F)-7 b(or)52 b(e)-5 b(ach)52 b Ft(")60 b(>)f Fy(0)52 b Fo(ther)-5 b(e)52 b(ar)-5 b(e)52 b Ft(u)60 b Fl(2)e Ft(L)2047 2799 y Fq(1)2087 2832 y Fy(\()p Ft(X)7 b Fy(\))52 b Fo(and)h Ft(v)62 b Fl(2)d Ft(L)2775 2799 y Fq(1)2814 2832 y Fy(\()p Ft(X)2931 2799 y FG(\003)2972 2832 y Fy(\))51 b Fo(such)h(that)448 2945 y Fl(k)p Ft(T)546 2959 y FF(u;v)668 2945 y Fl(\000)20 b Ft(T)13 b Fl(k)26 b Ft(<)f(")448 3058 y Fy(\(ii\))32 b Ft(T)38 b Fl(2)25 b Ft(C)851 3025 y Fq(0)844 3081 y(u)890 3058 y Fy(\(\004\))448 3171 y(\(iii\))31 b(lim)754 3185 y FF(x)p FG(!)p Fq(0)918 3171 y Fl(k)p Fy(\()p Fl(U)1055 3185 y FF(x)1120 3171 y Fl(\000)20 b Fy(1\))p Ft(T)13 b Fl(k)26 b Fy(=)f(0)33 b Fo(and)h Fy(lim)1905 3186 y FF(k)r FG(!)p Fq(0)2069 3171 y Fl(k)p Fy(\()p Fl(V)2205 3186 y FF(k)2268 3171 y Fl(\000)20 b Fy(1\))p Ft(T)13 b Fl(k)26 b Fy(=)f(0)p Fo(.)448 3284 y(Mor)-5 b(e)g(over,)33 b(if)f(one)g(of)g(these)g(c)-5 b(onditions)33 b(is)f(satis\014e)-5 b(d,)33 b(the)f(functions)g Ft(u)g Fo(and)h Ft(v)i Fo(fr)-5 b(om)448 3397 y Fy(\(i\))42 b Fo(may)h(b)-5 b(e)41 b(chosen)i(such)f (that)h(their)f(F)-7 b(ourier)43 b(tr)-5 b(ansforms)53 b Fm(b)-60 b Ft(u)16 b Fo(,)49 b Fm(b)-57 b Ft(v)61 b Fo(have)42 b(c)-5 b(omp)g(act)448 3510 y(supp)g(ort.)448 3685 y Fz(Pro)s(of:)48 b Fy(First,)30 b(\(ii\))f(is)h(equiv)-5 b(alen)m(t)29 b(to)j(\(iii\),)d(as)h(a)h(consequence)g(of)448 3875 y Fl(kW)583 3894 y Fq(\()p FF(x;k)r Fq(\))740 3875 y Fy([)p Ft(T)13 b Fy(])r Fl(\000)r Ft(T)g Fl(k)26 b Fy(=)f Fl(kU)1266 3889 y FF(x)1310 3875 y Fy(\()p Fl(V)1401 3890 y FF(k)1444 3875 y Fy([)p Ft(T)13 b Fy(])r Fl(\000)r Ft(T)g Fy(\))r(+)r(\()p Fl(U)1903 3889 y FF(x)1948 3875 y Fy([)p Ft(T)g Fy(])r Fl(\000)r Ft(T)g Fy(\))p Fl(k)26 b(\024)f(kV)2508 3890 y FF(k)2550 3875 y Fy([)p Ft(T)13 b Fy(])r Fl(\000)r Ft(T)g Fl(k)r Fy(+)r Fl(kU)3029 3889 y FF(x)3074 3875 y Fy([)p Ft(T)g Fy(])r Fl(\000)r Ft(T)g Fy(\))p Fl(k)p Ft(:)448 4065 y Fy(W)-8 b(e)30 b(pro)m(v)m(e)f(no)m(w)g (the)g(equiv)-5 b(alence)28 b(b)s(et)m(w)m(een)h(\(i\))f(and)g(\(ii\).) 40 b(F)-8 b(or)29 b(eac)m(h)h(couple)e(\()p Ft(y)s(;)15 b(p)p Fy(\))26 b Fl(2)448 4178 y Ft(X)i Fl(\002)20 b Ft(X)724 4145 y FG(\003)794 4178 y Fy(w)m(e)31 b(ha)m(v)m(e)553 4394 y Fl(W)643 4413 y Fq(\()p FF(y)r(;p)p Fq(\))795 4394 y Fy([)p Ft(T)873 4408 y FF(u;v)974 4394 y Fy(])26 b Fl(\021)f Fy(\()p Fl(U)1213 4408 y FF(y)1254 4394 y Fl(V)1310 4408 y FF(p)1350 4394 y Fy(\)[)p Ft(T)1463 4408 y FF(u;v)1565 4394 y Fy(])g(=)1711 4270 y Fm(Z)1762 4477 y FF(X)1844 4270 y Fm(Z)1895 4477 y FF(X)1958 4458 y Fx(\003)1998 4394 y Fy(\()p Fl(U)2090 4408 y FF(x)2134 4394 y Fl(V)2190 4409 y FF(k)2233 4394 y Fy(\)[)p Ft(T)13 b Fy(])i Ft(u)p Fy(\()p Ft(x)21 b Fl(\000)f Ft(y)s Fy(\))15 b Ft(v)s Fy(\()p Ft(k)24 b Fl(\000)c Ft(p)p Fy(\))15 b(d)p Ft(x)g Fy(d)p Ft(k)s(:)448 4638 y Fy(Since)48 b(the)g (translations)g(act)h(con)m(tin)m(uously)f(on)g Ft(L)2315 4605 y Fq(1)2403 4638 y Fy(w)m(e)h(deduce)f(that)h(the)f(map)448 4751 y(\()p Ft(y)s(;)15 b(p)p Fy(\))43 b Fl(7!)f(W)918 4769 y Fq(\()p FF(y)r(;p)p Fq(\))1070 4751 y Fy([)p Ft(T)1148 4765 y FF(u;v)1249 4751 y Fy(])f(is)f(norm)g(con)m(tin)m(uous.)71 b(So)40 b Ft(T)2375 4765 y FF(u;v)2519 4751 y Fl(2)i Ft(C)2694 4718 y Fq(0)2687 4773 y(u)2733 4751 y Fy(\(\004\))e(for)h (eac)m(h)g Ft(T)56 b Fl(2)448 4863 y Ft(B)5 b Fy(\()p Fn(H)27 b Fy(\).)58 b(Then)35 b(if)g(\(i\))h(holds)f(w)m(e)i(get)g (\(ii\))e(b)s(ecause)i Ft(C)2368 4830 y Fq(0)2361 4886 y(u)2406 4863 y Fy(\(\004\))g(is)e(a)i(norm)e(closed)h(sub-)448 4976 y(space)30 b(of)f Ft(B)5 b Fy(\()p Fn(H)26 b Fy(\).)41 b(Recipro)s(cally)-8 b(,)28 b(assume)h(that)g(\(iii\))f(holds.)39 b(It)29 b(can)g(b)s(e)f(sho)m(wn)g(that)1897 5225 y(53)p eop %%Page: 54 54 54 53 bop 448 573 a Fy(for)32 b(ev)m(ery)g(op)s(en)f(set)h(\003)c Fl(6)p Fy(=)f Fl(;)32 b Fy(of)f Ft(X)39 b Fy(and)31 b(for)h(ev)m(ery)g Ft(")c(>)f Fy(0)32 b(there)f(is)g Ft(u)c Fl(2)g Ft(L)3038 540 y Fq(1)3078 573 y Fy(\()p Ft(X)7 b Fy(\))32 b(suc)m(h)448 686 y(that)j Ft(u)d Fl(\025)f Fy(0,)941 613 y Fm(R)984 718 y FF(X)1066 686 y Ft(u)h Fy(=)f(1,)1358 613 y Fm(R)1401 718 y FF(X)5 b FG(n)p Fq(\003)1568 686 y Ft(u)32 b Fl(\024)f Ft(")j Fy(and)42 b Fm(b)-59 b Ft(u)47 b Fl(2)31 b Ft(C)2267 700 y Fq(c)2303 686 y Fy(\()p Ft(X)2420 653 y FG(\003)2460 686 y Fy(\))j(\(put)g Ft(u)e Fy(=)2935 662 y Fm(b)2921 686 y Ft( )53 b Fy(in)33 b(Lemma)448 811 y(2.1)41 b(from)e([GI1)q(]\).) 68 b(Similarly)-8 b(,)39 b(for)g(eac)m(h)i(neigh)m(b)s(ourho)s(o)s(d)36 b(\000)j(of)h(0)g(in)e Ft(X)3055 778 y FG(\003)3134 811 y Fy(there)i(is)448 924 y Ft(v)35 b Fl(2)d Ft(L)682 891 y Fq(1)721 924 y Fy(\()p Ft(X)838 891 y FG(\003)878 924 y Fy(\))j(with)e Ft(v)i Fl(\025)c Fy(0,)1446 850 y Fm(R)1489 956 y FF(X)1552 937 y Fx(\003)1608 924 y Ft(v)k Fy(=)c(1,)1895 850 y Fm(R)1938 956 y FF(X)2001 937 y Fx(\003)2037 956 y FG(n)p Fq(\000)2136 924 y Ft(v)k Fl(\024)c Ft(")k Fy(and)40 b Fm(b)-57 b Ft(v)50 b Fl(2)31 b Ft(C)2826 938 y Fq(c)2862 924 y Fy(\()p Ft(X)7 b Fy(\).)53 b(Then)34 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y Fy(\))36 b Fo(then)448 2536 y Ft(T)501 2550 y FF(u;v)628 2536 y Fl(2)25 b Fs(A)p Fo(.)448 2743 y Fz(Pro)s(of:)48 b Fy(The)34 b(condition)g(\(i\))h(of)g(Theorem)g(5.11)i(is)d(stronger)h (than)g(lim)2997 2757 y FF(x)p FG(!)p Fq(0)3161 2743 y Fl(k)p Fy(\()p Fl(U)3298 2757 y FF(x)3366 2743 y Fl(\000)448 2856 y Fy(1\))p Ft(T)13 b Fl(k)27 b Fy(=)e(0)g(from)f(\(iii\))f(of)i (Lemma)g(5.13,)i(hence)e(eac)m(h)h Ft(T)38 b Fl(2)25 b Fs(A)g Fy(v)m(eri\014es)f(the)h(statemen)m(ts)448 2969 y(\(i\)-\(iii\))33 b(of)h(this)f(lemma.)52 b(It)34 b(remains)f(to)h (pro)m(v)m(e)h(that)g Ft(T)2445 2983 y FF(u;v)2578 2969 y Fl(2)c Fs(A)p Fy(,)36 b(whic)m(h)c(ob)m(viously)448 3082 y(follo)m(ws)e(b)m(y)g(dominated)g(con)m(v)m(ergence)i(if)e Fl(W)1999 3097 y FF(\030)2036 3082 y Fy([)p Ft(T)13 b Fy(])26 b Fl(2)f Fs(A)30 b Fy(for)h(an)m(y)f Ft(\030)f Fl(2)c Ft(X)j Fl(\002)20 b Ft(X)3107 3049 y FG(\003)3147 3082 y Fy(.)589 3195 y(Let,)28 b(more)d(generally)-8 b(,)26 b Ft( )j Fy(b)s(e)c(the)g(F)-8 b(ourier)25 b(transform)g(of)g (an)g(in)m(tegrable)g(measure)448 3308 y(on)35 b Ft(X)43 b Fy(and)34 b Ft(T)46 b Fl(2)32 b Fs(A)p Fy(.)55 b(W)-8 b(e)36 b(shall)d(pro)m(v)m(e)j(that)g Ft( )s Fy(\()p Ft(P)13 b Fy(\))p Ft(T)46 b Fl(2)33 b Fs(A)p Fy(.)55 b(Since)34 b Ft(U)2902 3322 y FF(x)2979 3308 y Fy(=)e Ft(x)p Fy(\()p Ft(P)13 b Fy(\))36 b(the)448 3421 y(op)s(erator)31 b Ft( )s Fy(\()p Ft(P)13 b Fy(\))p Ft(T)44 b Fy(clearly)30 b(satis\014es)g(condition)f(\(i\))h(of)h(Theorem)f(5.11.)42 b(Then)867 3619 y([)p Ft(V)945 3634 y FF(k)988 3619 y Ft(;)15 b( )s Fy(\()p Ft(P)e Fy(\))p Ft(T)g Fy(])27 b(=)e Fl(f)p Ft(V)1543 3634 y FF(k)1586 3619 y Ft( )s Fy(\()p Ft(P)13 b Fy(\))p Ft(V)1864 3582 y FG(\003)1842 3643 y FF(k)1923 3619 y Fl(\000)20 b Ft( )s Fy(\()p Ft(P)13 b Fy(\))p Fl(g)p Ft(V)2315 3634 y FF(k)2359 3619 y Ft(T)33 b Fy(+)20 b Ft( )s Fy(\()p Ft(P)13 b Fy(\)[)p Ft(V)2817 3634 y FF(k)2861 3619 y Ft(;)i(T)e Fy(])p Ft(:)448 3818 y Fy(Since)36 b Ft(V)745 3833 y FF(k)788 3818 y Ft( )s Fy(\()p Ft(P)13 b 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b(j)2688 4916 y Fm(b)2674 4940 y Ft( )19 b Fl(j)p Fy(\()p Ft(dx)p Fy(\))p Ft(:)1897 5225 y Fy(54)p eop %%Page: 55 55 55 54 bop 448 573 a Fy(Since)43 b Ft(T)59 b Fl(2)46 b Fs(A)p Fy(,)h(there)d(is)e(a)h(compact)i(set)f Ft(M)53 b Fy(in)42 b Ft(X)50 b Fy(suc)m(h)43 b(that)h Fl(k)2889 555 y Ft(\037)2946 587 y FF(L)2994 564 y Fv(c)2994 610 y(M)3066 573 y Fy(\()p Ft(Q)p Fy(\))p Ft(T)13 b Fl(k)47 b(\024)448 708 y Ft(")p Fy([2)p Fl(j)599 684 y Fm(b)585 708 y Ft( )20 b Fl(j)p Fy(\()p Ft(K)7 b Fy(\)])868 675 y FG(\000)p Fq(1)964 708 y Fy(.)55 b(If)35 b(w)m(e)h(tak)m(e)h(\003)d (=)g Ft(M)f Fl(\000)24 b Ft(K)42 b Fy(then)35 b(\003)g(is)g(also)g(a)h (compact)h(in)d Ft(X)43 b Fy(and)448 821 y(for)33 b(eac)m(h)h Ft(x)c Fl(2)f Ft(K)39 b Fy(w)m(e)34 b(ha)m(v)m(e)g Ft(M)40 b Fl(\032)29 b Ft(x)22 b Fy(+)f(\003,)34 b(so)g Ft(L)2127 835 y FF(M)2235 821 y Fl(\032)29 b Ft(L)2397 835 y Fq(\003)2472 821 y Fy(+)22 b Ft(x)p Fy(.)48 b(But)2873 803 y Ft(\037)2930 835 y FF(L)2978 812 y Fv(c)2978 858 y Fk(\003)3028 821 y Fy(\()p Ft(Q)22 b Fl(\000)g Ft(x)p Fy(\))29 b(=)448 916 y Ft(\037)505 952 y Fq(\()p FF(L)580 963 y Fk(\003)626 952 y Fq(+)p FF(x)p Fq(\))748 933 y Fv(c)784 934 y Fy(\()p Ft(Q)p Fy(\))39 b(so)e Fl(k)1128 916 y Ft(\037)1186 948 y FF(L)1234 925 y Fv(c)1234 971 y Fk(\003)1283 934 y Fy(\()p Ft(Q)26 b Fl(\000)e Ft(x)p Fy(\))p Ft(T)13 b Fl(k)38 b(\024)f(k)1900 916 y Ft(\037)1958 948 y FF(L)2006 925 y Fv(c)2006 971 y(M)2077 934 y Fy(\()p Ft(Q)p Fy(\))p Ft(T)13 b Fl(k)p Fy(.)63 b(With)37 b(this)g(c)m(hoice)h(of)g(\003)g(w)m (e)448 1047 y(shall)29 b(th)m(us)h(ha)m(v)m(e)i Fl(k)1116 1029 y Ft(\037)1173 1061 y FF(L)1221 1038 y Fv(c)1221 1084 y Fk(\003)1271 1047 y Fy(\()p Ft(Q)p Fy(\))p Ft( )s Fy(\()p Ft(P)13 b Fy(\))p Ft(T)g Fl(k)27 b(\024)e Ft(")p Fy(.)p 3371 1039 67 67 v 448 1309 a Fz(Lemma)33 b(5.15)46 b Fo(L)-5 b(et)32 b Ft(T)1259 1323 y FF(u;v)1392 1309 y Fo(b)-5 b(e)31 b(given)g(by)h(\(5.11\))h(with)38 b Fm(b)-57 b Ft(v)50 b Fo(of)31 b(c)-5 b(omp)g(act)34 b(supp)-5 b(ort.)44 b(Then)448 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FF(x)1825 2284 y Ft(V)1898 2246 y FG(\003)1878 2307 y FF(k)1953 2284 y Ft(u)p Fy(\()p Ft(x)p Fy(\))i Ft(v)s Fy(\()p Ft(k)s Fy(\))32 b(d)p Ft(x)15 b Fy(d)o Ft(k)34 b(\022)2633 2298 y Fq(2)2672 2284 y Fy(\()p Ft(Q)p Fy(\))26 b(=)f(0)448 2529 y(for)38 b(supp)14 b Ft(\022)841 2543 y Fq(1)918 2529 y Fy(and)38 b(supp)13 b Ft(\022)1348 2543 y Fq(2)1426 2529 y Fy(su\016cien)m(tly)37 b(far)h(a)m(w)m(a)m(y)i (one)e(from)g(another)g(and)g(for)g(all)448 2642 y Ft(T)k Fl(2)28 b Ft(B)5 b Fy(\()p Fn(H)26 b Fy(\).)47 b(By)33 b(the)g(w)m(eak)g(densit)m(y)f(of)g(the)h(\014nite)e(rank)h(op)s (erators,)h(it)f(su\016ces)g(to)448 2755 y(assume)j Ft(T)47 b Fy(of)35 b(rank)f(one)g(and,)i(b)m(y)e(the)h(p)s(olarization)e(iden)m (tit)m(y)-8 b(,)35 b(w)m(e)g(ma)m(y)h(tak)m(e)g Ft(T)47 b Fy(of)448 2868 y(the)31 b(form)f Fl(j)p Ft(g)s Fl(ih)p Ft(g)s Fl(j)j Fy(for)d(some)g Ft(g)f Fl(2)c Fn(H)i Fy(.)40 b(Then)30 b(for)g(an)m(y)h Ft(f)2364 2882 y Fq(1)2403 2868 y Ft(;)f(f)2503 2882 y Fq(2)2567 2868 y Fl(2)25 b Fn(H)57 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Fq(1)1090 4035 y Fy(d)p Ft(x)1193 4049 y Fq(2)1247 4035 y Ft(u)p Fy(\()p Ft(x)p Fy(\))p 1436 3961 84 4 v 15 w Ft(f)1481 4049 y Fq(1)1521 4035 y Fy(\()p Ft(x)1608 4049 y Fq(1)1648 4035 y Fy(\))g Ft(\022)1741 4049 y Fq(1)1781 4035 y Fy(\()p Ft(x)1868 4049 y Fq(1)1907 4035 y Fy(\))g Ft(g)s Fy(\()p Ft(x)2090 4049 y Fq(1)2151 4035 y Fy(+)20 b Ft(x)p Fy(\))p 2344 3985 47 4 v 15 w Ft(g)t Fy(\()p Ft(x)2478 4049 y Fq(2)2538 4035 y Fy(+)g Ft(x)p Fy(\))15 b Ft(\022)2774 4049 y Fq(2)2814 4035 y Fy(\()p Ft(x)2901 4049 y Fq(2)2941 4035 y Fy(\))g Ft(f)3036 4049 y Fq(2)3075 4035 y Fy(\()p Ft(x)3162 4049 y Fq(2)3202 4035 y Fy(\))5 b Fl(\002)642 4292 y(\002)728 4169 y Fm(Z)778 4389 y Fv(X)832 4375 y Fx(\003)844 4292 y Ft(k)s Fy(\()p Ft(x)981 4306 y Fq(1)1041 4292 y Fl(\000)20 b Ft(x)1184 4306 y Fq(2)1224 4292 y Fy(\))15 b Ft(v)s Fy(\()p Ft(k)s Fy(\))g(d)q Ft(k)s(:)448 4565 y Fy(The)36 b(last)g(in)m(tegral)g(o)m(v)m(er)i Ft(X)1439 4532 y FG(\003)1515 4565 y Fy(equals)e(\()p Fl(F)1906 4532 y FG(\000)p 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67 67 v 1897 5225 a(55)p eop %%Page: 56 56 56 55 bop 589 573 a Fy(As)29 b(a)f(consequence)h(of)g(the)f(previous)f (results,)h(for)g(eac)m(h)h Ft(T)38 b Fl(2)25 b Fs(A)k Fy(and)f(eac)m(h)h Ft(")d(>)f Fy(0)448 686 y(there)i(are)g Ft(u)e Fl(2)g Ft(L)1050 653 y Fq(1)1089 686 y Fy(\()p Ft(X)7 b Fy(\))28 b(and)e Ft(v)i Fl(2)d Ft(L)1662 653 y Fq(1)1702 686 y Fy(\()p Ft(X)1819 653 y FG(\003)1859 686 y Fy(\))i(with)k Fm(b)-57 b Ft(v)45 b Fy(compactly)27 b(supp)s(orted)d(suc)m(h)i(that)448 799 y Ft(T)501 813 y FF(u;v)636 799 y Fl(2)34 b Fs(A)p Fy(,)j Fl(k)p Ft(T)962 813 y FF(u;v)1087 799 y Fl(\000)24 b Ft(T)13 b Fl(k)33 b Ft(<)h(")h Fy(and)g(suc)m(h)g(that)h(the)g(conclusion)e(of)h(Lemma)h (5.15)h(b)s(e)448 912 y(satis\014ed.)i(Hence)29 b(there)e(is)g(\(and)g (w)m(e)h(ma)m(y)g(\014x)f(it\))g(a)h(compact)h(set)f(\003)d Fl(\032)g Ft(X)35 b Fy(suc)m(h)27 b(that)448 1024 y(Lemma)k(5.15)h(b)s (e)d(v)-5 b(alid)29 b(and)h(for)g(whic)m(h)f(w)m(e)i(ha)m(v)m(e)448 1217 y(\(5.12\))572 b Fl(k)1295 1199 y Ft(\037)1353 1231 y FF(L)1401 1208 y Fv(c)1401 1254 y Fk(\003)1450 1217 y Fy(\()p Ft(Q)p Fy(\))p Ft(T)1645 1231 y FF(u;v)1747 1217 y Fl(k)21 b Fy(+)f Fl(k)p Ft(T)2002 1231 y FF(u;v)2104 1199 y Ft(\037)2161 1231 y FF(L)2209 1208 y Fv(c)2209 1254 y Fk(\003)2258 1217 y Fy(\()p Ft(Q)p Fy(\))p Fl(k)27 b Ft(<)e(":)448 1410 y Fy(Let)f(no)m(w)f Ft(\022)k Fl(2)e Ft(C)1007 1424 y Fq(c)1043 1410 y Fy(\()p Ft(X)7 b Fy(\))24 b(with)e(0)j Fl(\024)g Ft(\022)j Fl(\024)d Fy(1)e(and)g Ft(\022)k Fy(=)e(1)f(on)f(\003.)38 b(F)-8 b(or)24 b(eac)m(h)g Ft(l)j Fl(2)e Ft(L)e Fy(w)m(e)g(denote)448 1522 y(simply)29 b(b)m(y)i Ft(\022)913 1490 y FF(l)969 1522 y Fy(b)s(oth)f(the)h(map)g Ft(x)26 b Fl(7!)g Ft(\022)s Fy(\()p Ft(x)20 b Fl(\000)g Ft(l)r Fy(\))31 b(and)g(the)g(op)s(erator)g(of)g(m)m(ultiplication)448 1635 y(b)m(y)41 b(this)e(function)g(\(in)h(what)g(follo)m(ws)g(w)m(e)h (shall)e(freely)h(use)g(the)h(same)g(condensed)448 1748 y(notation)35 b(for)g(other)f(functions)f(on)i Ft(X)42 b Fy(to)s(o\).)54 b(Since)34 b(the)h(set)g Ft(L)f Fy(is)g(sparse,)h(w)m (e)g(ma)m(y)448 1861 y(\014nd)29 b(a)i(subset)f Ft(M)35 b Fl(\032)25 b Ft(L)30 b Fy(with)f(a)i(\014nite)e(complemen)m(tary)i (suc)m(h)f(that)917 2054 y(supp)13 b Ft(\022)1165 2016 y FF(l)1205 1986 y Fm(T)1306 2054 y Fy(\(supp)h Ft(\022)1590 2021 y FF(l)1612 1997 y Fx(0)1658 2054 y Fy(+)20 b(\003\))26 b(=)f Fl(;)61 b Fy(if)e Ft(l)r(;)31 b(l)2302 2021 y FG(0)2350 2054 y Fl(2)25 b Ft(M)41 b Fy(and)29 b Ft(l)e Fl(6)p Fy(=)e Ft(l)2920 2021 y FG(0)2944 2054 y Ft(:)448 2246 y Fy(Lemma)34 b(5.15)i(giv)m(es)e(then)f Ft(\022)1454 2213 y FF(l)1480 2246 y Ft(T)1533 2260 y FF(u;v)1634 2246 y Ft(\022)1680 2213 y FF(l)1702 2190 y Fx(0)1759 2246 y Fy(=)d(0)35 b(for)e(all)g(the)h(pairs)e Ft(l)r(;)15 b(l)2696 2213 y FG(0)2754 2246 y Fy(as)34 b(ab)s(o)m(v)m(e.)52 b(Hence,)448 2359 y(setting)31 b Ft(\036)25 b Fy(=)922 2291 y Fm(P)1018 2386 y FF(l)q FG(2)p FF(M)1181 2359 y Ft(\022)1227 2326 y FF(l)1283 2359 y Fy(one)30 b(obtains)1163 2574 y Ft(\036)15 b(T)1285 2588 y FF(u;v)1402 2574 y Ft(\036)26 b Fy(=)1616 2487 y Fm(X)1578 2685 y FF(l)q(;l)1642 2667 y Fx(0)1663 2685 y FG(2)p FF(M)1800 2574 y Ft(\022)1846 2536 y FF(l)1872 2574 y Ft(T)1925 2588 y FF(u;v)2026 2574 y Ft(\022)2072 2536 y FF(l)2094 2513 y Fx(0)2145 2574 y Fy(=)2247 2487 y Fm(X)2241 2685 y FF(l)q FG(2)p FF(M)2400 2574 y Ft(\022)2446 2536 y FF(l)2471 2574 y Ft(T)2524 2588 y FF(u;v)2626 2574 y Ft(\022)2672 2536 y FF(l)2697 2574 y Ft(:)448 2858 y Fy(On)c(the)h(other)g(hand,)h(since) e(0)k Fl(\024)f Ft(\036)g Fl(\024)g Fy(1)e(there)g(is)f(a)h(b)s (ounded,)g(compactly)g(supp)s(orted)448 2971 y(function)41 b Ft(')h Fy(suc)m(h)f(that)h(1)29 b Fl(\000)e Ft(\036)44 b Fy(=)g Ft(')28 b Fy(+)g(\(1)h Fl(\000)e Ft(\036)p Fy(\))2208 2954 y Ft(\037)2265 2985 y FF(L)2313 2963 y Fv(c)2313 3009 y Fk(\003)2363 2971 y Fy(.)75 b(This)40 b(giv)m(es)i(the)g(follo)m (wing)448 3084 y(decomp)s(osition)29 b(of)i Ft(T)1204 3098 y FF(u;v)1305 3084 y Fy(:)460 3277 y Ft(T)513 3291 y FF(u;v)640 3277 y Fy(=)25 b Ft(\036)15 b(T)858 3291 y FF(u;v)975 3277 y Ft(\036)20 b Fy(+)g(\(1)h Fl(\000)f Ft(\036)p Fy(\))p Ft(T)1474 3291 y FF(u;v)1591 3277 y Ft(\036)g Fy(+)g Ft(T)1809 3291 y FF(u;v)1911 3277 y Fy(\(1)h Fl(\000)f Ft(\036)p Fy(\))498 3432 y(=)612 3346 y Fm(X)606 3543 y FF(l)q FG(2)p FF(M)765 3432 y Ft(\022)811 3395 y FF(l)836 3432 y Ft(T)889 3446 y FF(u;v)990 3432 y Ft(\022)1036 3395 y FF(l)1082 3432 y Fy(+)g(\()p Ft(')15 b(T)1335 3446 y FF(u;v)1452 3432 y Ft(\036)21 b Fy(+)f Ft(T)1671 3446 y FF(u;v)1787 3432 y Ft(')p Fy(\))h(+)f(\(\(1)i Fl(\000)e Ft(\036)p Fy(\))2310 3414 y Ft(\037)2367 3446 y FF(L)2415 3423 y Fv(c)2415 3469 y Fk(\003)2465 3432 y Ft(T)2518 3446 y FF(u;v)2634 3432 y Ft(\036)h Fy(+)f Ft(T)2853 3446 y FF(u;v)2954 3414 y Ft(\037)3011 3446 y FF(L)3059 3423 y Fv(c)3059 3469 y Fk(\003)3109 3432 y Fy(\(1)h Fl(\000)f Ft(\036)p Fy(\)\))589 3713 y(Let)41 b(us)f(observ)m(e)h(that)g(for)f(eac)m(h)i Ft(S)47 b Fl(2)42 b Fs(A)f Fy(and)e(eac)m(h)j(b)s(ounded)c(function)h(with)448 3826 y(compact)k(supp)s(ort)e Ft(')h Fy(on)g Ft(X)49 b Fy(the)42 b(op)s(erators)g Ft(')p Fy(\()p Ft(Q)p Fy(\))p Ft(S)48 b Fy(and)41 b Ft(S)5 b(')p Fy(\()p Ft(Q)p Fy(\))43 b(are)g(compact.)448 3939 y(Indeed,)34 b(c)m(ho)s(ose)g Ft(\036)c Fl(2)g Ft(C)1304 3953 y Fq(c)1339 3939 y Fy(\()p Ft(X)7 b Fy(\))35 b(suc)m(h)e(that)h Ft('\036)c Fy(=)g Ft(')p Fy(.)50 b(It)33 b(su\016ces)g(th)m(us)g(to)h(sho)m(w)f(that)448 4052 y Ft(\036)p Fy(\()p Ft(Q)p Fy(\))p Ft(S)j Fy(is)30 b(compact.)42 b(The)30 b(second)g(mem)m(b)s(er)g(of)g(the)h(estimate) 589 4245 y Fl(k)p Fy(\()p Ft(U)731 4259 y FF(x)796 4245 y Fl(\000)20 b Fy(1\))p Ft(\036)p Fy(\()p Ft(Q)p Fy(\))p Ft(S)5 b Fl(k)27 b(\024)e(k)p Ft(\036)p Fy(\()p Ft(Q)p Fy(\))p Fl(k)15 b(k)p Fy(\()p Ft(U)1835 4259 y FF(x)1902 4245 y Fl(\000)20 b Fy(1\))p Ft(S)5 b Fl(k)21 b Fy(+)f Fl(k)p Ft(U)2398 4259 y FF(x)2443 4245 y Ft(\036)p Fy(\()p Ft(Q)p Fy(\))p Ft(U)2711 4207 y FG(\003)2701 4267 y FF(x)2771 4245 y Fl(\000)g Ft(\036)p Fy(\()p Ft(Q)p Fy(\))p Fl(k)15 b(k)p Ft(S)5 b Fl(k)p Ft(;)448 4437 y Fy(tends)25 b(to)h(zero)g(b)s (ecause)f Ft(S)30 b Fl(2)25 b Fs(A)g Fy(and)g Ft(\036)g Fy(is)f(uniformly)e(con)m(tin)m(uous.)38 b(This)24 b(sho)m(ws)g(that) 448 4550 y Ft(\036)p Fy(\()p Ft(Q)p Fy(\))p Ft(S)36 b Fy(satis\014es)30 b(the)h(h)m(yp)s(othesis)e(of)h(the)h(compacit)m(y)g (criterion)e(Prop)s(osition)g(3.13.)589 4663 y(So)35 b Ft(')15 b(T)847 4677 y FF(u;v)964 4663 y Ft(\036)23 b Fy(+)f Ft(T)1187 4677 y FF(u;v)1304 4663 y Ft(')34 b Fy(is)g(a)g(compact)i(op)s(erator)e Ft(K)7 b Fy(.)52 b(Th)m(us)33 b(w)m(e)i(ma)m(y)g(use)f(\(5.12\))448 4776 y(to)d(get)1345 4897 y Fl(k)p Ft(T)1443 4911 y FF(u;v)1565 4897 y Fl(\000)1662 4811 y Fm(X)1656 5009 y FF(l)q FG(2)p FF(M)1815 4897 y Ft(\022)1861 4860 y FF(l)1886 4897 y Ft(T)1939 4911 y FF(u;v)2040 4897 y Ft(\022)2086 4860 y FF(l)2132 4897 y Fl(\000)20 b Ft(K)7 b Fl(k)25 b Ft(<)g(":)1897 5225 y Fy(56)p eop %%Page: 57 57 57 56 bop 589 573 a Fy(In)71 b(this)g(manner,)81 b(w)m(e)72 b(are)g(reduced)f(to)h(the)g(pro)s(of)f(of)h(the)f(assertion)448 618 y Fm(P)544 713 y FF(l)q FG(2)p FF(M)707 686 y Ft(\022)753 653 y FF(l)779 686 y Ft(T)832 700 y FF(u;v)933 686 y Ft(\022)979 653 y FF(l)1034 686 y Fl(2)30 b Fs(C)1186 700 y FF(L;)p Fq(0)1293 686 y Fy(.)49 b(This)31 b(ma)m(y)j(b)s(e)e (reform)m(ulated)h(as)2557 618 y Fm(P)2653 713 y FF(l)q FG(2)p FF(M)2816 686 y Ft(U)2888 653 y FG(\003)2878 713 y FF(l)2942 686 y Ft(K)3019 701 y FF(l)3061 686 y Ft(U)3123 701 y FF(l)3179 686 y Fl(2)c Fs(C)3330 700 y FF(L;)p Fq(0)448 799 y Fy(if)h(w)m(e)i(tak)m(e)h(in)m(to)e(accoun)m(t)i(that)f Ft(\022)1631 766 y FF(l)1682 799 y Fl(\021)27 b Ft(\022)s Fy(\()p Ft(Q)20 b Fl(\000)g Ft(l)r Fy(\))27 b(=)g Ft(U)2305 766 y FG(\003)2295 826 y FF(l)2344 799 y Ft(\022)s Fy(\()p Ft(Q)p Fy(\))p Ft(U)2594 814 y FF(l)2651 799 y Fy(and)k(if)f(w)m(e)h (denote)h(b)m(y)448 912 y Ft(K)525 927 y FF(l)583 912 y Fy(the)h(compact)g(op)s(erator)f Ft(\022)s Fy(\()p Ft(Q)p Fy(\))p Ft(U)1725 927 y FF(l)1751 912 y Ft(T)1804 926 y FF(u;v)1905 912 y Ft(U)1977 879 y FG(\003)1967 939 y FF(l)2017 912 y Ft(\022)s Fy(\()p Ft(Q)p Fy(\).)45 b(It)32 b(is)f(straigh)m(tforw)m(ard)h(to)g(c)m(hec)m(k)448 1024 y(that)d(the)g(family)e(of)i(these)f(compacts)i(v)m(eri\014es)e (the)h(h)m(yp)s(otheses)f(of)g(the)h(Prop)s(osition)448 1137 y(5.9,)40 b(cf.)61 b(the)37 b(Remark)g(after)h(that)g(prop)s (osition.)58 b(This)35 b(\014nishes)g(the)i(pro)s(of)g(of)g(the)448 1250 y(inclusion)28 b Fs(A)d Fl(\032)g Fs(C)1081 1264 y FF(L;)p Fq(0)1188 1250 y Fy(,)31 b(hence)f(that)h(of)g(Theorem)f (5.11.)448 1442 y Fz(5.5.)134 b Fy(In)39 b(this)f(paragraph)g(w)m(e)i (shall)d(p)s(oin)m(t)h(out)i(a)f(large)g(class)g(of)h(hamiltonians)448 1554 y(a\016liated)34 b(to)h(the)f(algebra)g Fs(C)1484 1568 y FF(L)1536 1554 y Fy(.)52 b(Although)33 b(w)m(e)i(consider)e (explicitly)e(only)j(the)g(case)448 1667 y Ft(X)46 b Fy(=)37 b Fp(R)737 1634 y FF(n)790 1667 y Fy(,)j(our)e(argumen)m(ts)g (easily)f(extend)h(to)h(other)f(groups)f(\(the)h(case)h Ft(X)46 b Fy(=)37 b Fp(Z)3394 1634 y FF(n)448 1780 y Fy(b)s(eing)29 b(m)m(uc)m(h)i(easier\).)589 1893 y(W)-8 b(e)36 b(b)s(egin)d(with)g(a)i(remark)f(concerning)g(the)h (de\014nition)d(of)j(the)f(hamiltonians.)448 2006 y(Let)42 b Fn(H)738 1973 y FF(s)817 2006 y Fy(=)h Fn(H)1047 1973 y FF(s)1084 2006 y Fy(\()p Fp(R)1179 1973 y FF(n)1232 2006 y Fy(\))e(b)s(e)g(the)g(scale)g(of)g(Sob)s(olev)g(spaces;)47 b(here)40 b Ft(s)j Fl(2)g Fp(R)s Ft(;)64 b Fn(H)3284 1973 y Fq(0)3366 2006 y Fy(=)448 2119 y Ft(L)510 2086 y Fq(2)550 2119 y Fy(\()p Fp(R)645 2086 y FF(n)698 2119 y Fy(\))33 b Fl(\021)f Fn(H)27 b Fy(.)54 b(Let)35 b Ft(s;)15 b(t)35 b Fy(b)s(e)f(real)h(n)m(um)m(b)s(ers)e(suc)m(h)i(that)g(0)e Fl(\024)g Ft(t)f(<)h(s)p Fy(.)53 b(Let)36 b Ft(H)3190 2133 y Fq(0)3263 2119 y Fy(b)s(e)f(a)448 2232 y(self-adjoin)m(t)30 b(op)s(erator)g(in)f Fn(H)57 b Fy(with)29 b Ft(D)s Fy(\()p Fl(j)p Ft(H)1955 2246 y Fq(0)1994 2232 y Fl(j)2019 2199 y Fq(1)p 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b(self-adjoin)m(t)g(op)s(erators)g (from)g(the)h(next)448 2796 y(theorem)f(should)d(b)s(e)i(in)m (terpreted)g(in)f(this)g(sense.)448 2979 y Fz(Theorem)34 b(5.16)47 b Fo(L)-5 b(et)32 b Ft(h)26 b Fy(:)g Fp(R)1469 2946 y FF(n)1547 2979 y Fl(!)f Fp(R)41 b Fo(b)-5 b(e)33 b(a)g(c)-5 b(ontinuous)34 b(function)e(such)h(that)448 3177 y Fy(\(5.13\))489 b Ft(C)1239 3139 y FG(\000)p Fq(1)1333 3177 y Fl(j)p Ft(x)p Fl(j)1435 3139 y Fq(2)p FF(s)1533 3177 y Fl(\024)25 b(j)p Ft(h)p Fy(\()p Ft(x)p Fy(\))p Fl(j)i(\024)e Ft(C)7 b Fl(j)p Ft(x)p Fl(j)2150 3139 y Fq(2)p FF(s)2254 3177 y Fo(for)33 b Fl(j)p Ft(x)p Fl(j)26 b Ft(>)f(R)q(;)448 3375 y Fo(for)43 b(some)f(c)-5 b(onstants)44 b Ft(s)d(>)g Fy(0)p Fo(,)j Ft(C)49 b(>)41 b Fy(0)h Fo(and)h Ft(R)f(<)f Fl(1)p Fo(;)46 b(denote)d Ft(H)2829 3389 y Fq(0)2909 3375 y Fy(=)f Ft(h)p Fy(\()p Ft(P)13 b Fy(\))p Fo(.)70 b(L)-5 b(et)448 3488 y Ft(t)27 b Fl(2)f Fy([0)p Ft(;)15 b(s)p Fy(\))34 b Fo(r)-5 b(e)g(al,)35 b(let)f Ft(L)f Fo(b)-5 b(e)33 b(a)h(sp)-5 b(arse)35 b(subset)e(of)h Fp(R)2149 3455 y FF(n)2202 3488 y Fo(,)f(and)i(let)e Fl(f)p Ft(V)2667 3503 y FF(l)2694 3488 y Fl(g)2739 3503 y FF(l)q FG(2)p FF(L)2893 3488 y Fo(b)-5 b(e)34 b(a)g(family)g(of)448 3601 y(symmetric)e(op)-5 b(er)g(ators)34 b(in)c Ft(B)5 b Fy(\()p Fn(H)1615 3568 y FF(t)1645 3601 y Ft(;)15 b Fn(H)1801 3568 y FG(\000)p FF(t)1886 3601 y Fy(\))30 b Fo(with)i(the)f(fol)5 b(lowing)31 b(pr)-5 b(op)g(erty:)44 b(ther)-5 b(e)31 b(is)g(a)448 3713 y(numb)-5 b(er)33 b Ft(a)26 b(>)f Fy(2)p Ft(n)32 b Fo(such)h(that)448 3911 y Fy(\(5.14\))681 b(sup)1369 3990 y FF(l)q FG(2)p FF(L)1511 3911 y Fl(kh)p Ft(Q)p Fl(i)1698 3874 y FF(a)1740 3911 y Ft(V)1793 3926 y FF(l)1819 3911 y Fl(k)1864 3931 y FF(B)s Fq(\()p Ff(H)2042 3912 y Fv(t)2070 3931 y FF(;)p Ff(H)2182 3912 y Fx(\000)p Fv(t)2258 3931 y Fq(\))2315 3911 y Ft(<)25 b Fl(1)p Ft(:)448 4176 y Fo(Then)66 b(the)f(series)1190 4108 y Fm(P)1286 4203 y FF(l)q FG(2)p FF(L)1422 4176 y Fy(e)1463 4143 y FG(\000)p FF(il)q(P)1622 4176 y Ft(V)1675 4191 y FF(l)1701 4176 y Fy(e)1742 4143 y FF(il)q(P)1911 4176 y Fo(c)-5 b(onver)g(ges)66 b(in)f(the)g(str)-5 b(ong)67 b(top)-5 b(olo)g(gy)68 b(of)448 4289 y Ft(B)5 b Fy(\()p Fn(H)673 4256 y FF(t)702 4289 y Ft(;)15 b Fn(H)858 4256 y FG(\000)p FF(t)943 4289 y Fy(\))40 b Fo(and)g(its)f(sum)h(is)f(a)h (symmetric)g(op)-5 b(er)g(ator)43 b Ft(V)57 b Fy(:)38 b Fn(H)2827 4256 y FF(t)2894 4289 y Fl(!)g Fn(H)3138 4256 y FG(\000)p FF(t)3223 4289 y Fo(.)62 b(L)-5 b(et)448 4402 y Ft(H)40 b Fy(=)33 b Ft(H)744 4416 y Fq(0)807 4402 y Fy(+)23 b Ft(V)d Fo(,)38 b Ft(H)1116 4417 y FF(l)1175 4402 y Fy(=)33 b Ft(H)1355 4416 y Fq(0)1417 4402 y Fy(+)23 b Ft(V)1564 4417 y FF(l)1627 4402 y Fo(b)-5 b(e)37 b(the)h (self-adjoint)g(op)-5 b(er)g(ators)40 b(in)d Fn(H)63 b Fo(de\014ne)-5 b(d)38 b(as)448 4515 y(form)j(sums.)64 b(Then)40 b Ft(H)46 b Fo(is)40 b(strictly)h(a\016liate)-5 b(d)41 b(to)g Fs(C)2303 4529 y FF(L)2355 4515 y Fo(,)g Ft(H)2500 4530 y FF(l)2565 4515 y Fo(is)f(strictly)h(a\016liate)-5 b(d)41 b(to)448 4628 y Fp(T)p Fy(\()p Ft(X)7 b Fy(\))p Fo(,)36 b(and)d(the)g(family)h Fl(f)p Ft(H)1448 4643 y FF(l)1474 4628 y Fl(g)1519 4643 y FF(l)q FG(2)p FF(L)1673 4628 y Fo(is)e(a)h(r)-5 b(epr)g(esentative)35 b(of)e Ft(H)7 b Fo(.)41 b(In)33 b(p)-5 b(articular:)1394 4858 y Fr(\033)1448 4872 y Fq(ess)1540 4858 y Fy(\()p Ft(H)7 b Fy(\))25 b(=)1856 4771 y Fm(\\)1838 4965 y Fv(F)8 b Fx(\032)p Fv(L)1814 5013 y(F)g Fk(\014nite)p 2014 4751 453 4 v 2066 4771 a Fm([)2014 4973 y FF(l)q FG(2)p FF(L)p FG(n)p FF(F)2235 4858 y Fr(\033)m Fy(\()p Ft(H)2405 4873 y FF(l)2430 4858 y Fy(\))q Ft(:)1897 5225 y Fy(57)p eop %%Page: 58 58 58 57 bop 448 573 a Fo(If)34 b Fj({)j Fo(is)c(an)i(ultr)-5 b(a\014lter)35 b(on)f Ft(L)g Fo(\014ner)g(than)h(the)f(F)-7 b(r)n(\023)-44 b(echet)34 b(\014lter,)h(then)f Fy(u)15 b(-lim)3074 588 y FF(l)q FG(2)p Fi({)3212 573 y Ft(H)3288 588 y FF(l)3341 573 y Fy(:=)448 686 y Ft(H)524 700 y Fi({)609 686 y Fo(exists)30 b(in)h(the)g(norm)h(r)-5 b(esolvent)31 b(sense,)g(one)g(has)h Ft(H)2407 700 y Fi({)2486 686 y Fy(=)24 b(s)16 b(-lim)2789 701 y FF(l)q(;)p Fi({)2900 686 y Fy(e)2940 653 y FF(il)q(P)3045 686 y Ft(H)7 b Fy(e)3168 653 y FG(\000)p FF(il)q(P)3358 686 y Fo(in)448 799 y(the)33 b(str)-5 b(ong)34 b(r)-5 b(esolvent)34 b(sense,)f(and)1505 1011 y Fr(\033)1559 1025 y Fq(ess)1650 1011 y Fy(\()p Ft(H)7 b Fy(\))26 b(=)1953 925 y Fm([)1925 1141 y Fi({)s FG(2)2032 1125 y Fh(e)2022 1141 y FF(L)2097 1011 y Fr(\033)l Fy(\()p Ft(H)2266 1025 y Fi({)2320 1011 y Fy(\))p Ft(:)448 1322 y Fz(Remarks:)47 b(\(i\))35 b Fy(If)g Ft(s)f Fl(\024)g Ft(n=)p Fy(2)i(and)f Ft(V)1768 1337 y FF(l)1828 1322 y Fy(:)f Fp(R)1947 1289 y FF(n)2034 1322 y Fl(!)h Fp(R)44 b Fy(are)36 b(Borel)g(functions)e(satisfying)448 1435 y(the)d(condition)1004 1362 y Fm(R)1047 1467 y FG(j)p FF(y)r FG(\000)p FF(x)p FG(j)p FF(<)p Fq(1)1328 1435 y Fl(j)p Ft(V)1406 1450 y FF(l)1432 1435 y Fy(\()p Ft(y)s Fy(\))p Fl(j)21 b(\001)g(j)p Ft(y)i Fl(\000)d Ft(x)p Fl(j)1903 1402 y FG(\000)p FF(n)p Fq(+2)p FF(s)p FG(\000)p FF(\025)2239 1435 y Fy(d)o Ft(y)29 b Fl(\024)c Ft(c)p Fl(h)p Ft(x)p Fl(i)2620 1402 y FG(\000)p FF(a)2743 1435 y Fl(8)p Ft(x)g Fl(2)g Fp(R)3017 1402 y FF(n)3100 1435 y Fy(for)31 b(some)448 1548 y(constans)k Ft(c;)15 b(\025)33 b(>)f Fy(0,)37 b(then)d(the)h(op)s(erators)f Ft(V)2028 1563 y FF(l)2089 1548 y Fy(of)h(m)m(ultiplication)c(b)m(y)k(the)f (functions)448 1661 y Ft(V)501 1676 y FF(l)569 1661 y Fy(satisfy)41 b(\(5.14\))j(for)d(some)h Ft(t)i(<)g(s)p Fy(.)74 b(If)41 b Ft(s)j(>)g(n=)p Fy(2)e(then)f(the)h(simpler)d (condition)448 1701 y Fm(R)491 1806 y FG(j)p FF(y)r FG(\000)p FF(x)p FG(j)p FF(<)p Fq(1)772 1774 y Fl(j)p Ft(V)850 1789 y FF(l)876 1774 y Fy(\()p Ft(y)s Fy(\))p Fl(j)15 b Fy(d)q Ft(y)28 b Fl(\024)d Ft(c)p Fl(h)p Ft(x)p Fl(i)1416 1741 y FG(\000)p FF(a)1542 1774 y Fy(su\016ces.)40 b(If)29 b Ft(s)g Fy(is)g(an)g(in)m(teger)h(and)e Ft(V)2792 1789 y FF(l)2848 1774 y Fy(is)g(a)i(di\013eren)m(tial)448 1887 y(op)s(erator)j(of)f(order)g(less)f(than)h(2)p Ft(s)p 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Ft(L)755 2757 y Fq(2)794 2790 y Fy(\()p Ft(X)7 b Fy(\))33 b(of)f(ph)m(ysical)e (states)j(b)m(y)e Ft(L)1883 2757 y Fq(2)1923 2790 y Fy(\()p Ft(X)7 b Fy(;)15 b Fg(E)p Fy(\))33 b(where)e Fg(E)g Fy(is)g(a)h (\014nite)f(dimensional)448 2903 y(Hilb)s(ert)f(space.)46 b(This)30 b(allo)m(ws)h(us)h(to)g(treat,)i(for)d(example,)i(Dirac)f (hamiltonians)d Ft(H)3398 2917 y Fq(0)448 3016 y Fy(p)s(erturb)s(ed)21 b(b)m(y)i(the)g(same)h(class)f(of)h(p)s(oten)m(tials)e Ft(V)e Fy(.)39 b(The)22 b(condition)g(\(5.13\))k(is)c(satis\014ed)448 3129 y(with)36 b Ft(s)h Fy(=)g(1)p Ft(=)p Fy(2,)j Fl(j)p Ft(h)p Fy(\()p Ft(x)p Fy(\))p Fl(j)f Fy(b)s(eing)d(in)m(terpreted)h(as) g([)p Ft(h)p Fy(\()p Ft(x)p Fy(\))2357 3096 y Fq(2)2398 3129 y Fy(])2423 3096 y Fq(1)p FF(=)p Fq(2)2571 3129 y Fy(\()p Ft(h)p Fy(\()p Ft(x)p Fy(\))i(is)d(a)i(self-adjoin)m(t)448 3242 y(op)s(erator)f(in)d Fg(E)p Fy(\).)58 b(Note)38 b(that)e Ft(H)1614 3256 y Fq(0)1689 3242 y Fy(is)f(not)i(semib)s (ounded)c(in)i(this)g(case.)58 b(Theorems)448 3355 y(5.6)32 b(and)e(5.8)h(remain)e(v)-5 b(alid)29 b(without)g(an)m(y)i(c)m(hange)h (in)d(this)g(con)m(text.)448 3468 y Fz(\(iv\))35 b Fy(The)f(assumption) g Ft(t)e(<)h(s)i Fy(is)f(not)h(essen)m(tial)g(and)f(can)h(b)s(e)g (impro)m(v)m(ed)f(to)i Ft(t)d Fy(=)f Ft(s)p Fy(,)448 3581 y(whic)m(h)e(allo)m(ws)h(one)g(to)h(treat)h(p)s(erturbations)c(of) i(the)h(same)f(order)g(as)h Ft(H)2972 3595 y Fq(0)3011 3581 y Fy(.)43 b(But)31 b(then)448 3694 y(one)g(m)m(ust)f(add)f(other)i (conditions)d(in)h(order)h(to)h(giv)m(e)f(a)h(sense)f(to)h(the)f(sums)f Ft(H)3209 3708 y Fq(0)3268 3694 y Fy(+)19 b Ft(V)3411 3709 y FF(l)448 3806 y Fy(and)26 b Ft(H)697 3820 y Fq(0)748 3806 y Fy(+)12 b Ft(V)47 b Fy(as)26 b(self-adjoin)m(t)g(op)s(erators.) 40 b(This)24 b(question)i(is)f(of)i(some)g(imp)s(ortance)e(if)448 3919 y(one)i(w)m(an)m(ts)g(to)g(treat)h(Dirac)e(op)s(erators)h(with)e (Coulom)m(b)g(p)s(oten)m(tials)h(or)g(second)h(order)448 4032 y(p)s(erturbations)g(of)i(the)g(Laplace)g(op)s(erator,)h(but)e(is) g(outside)g(the)h(main)e(scop)s(e)i(of)g(this)448 4145 y(pap)s(er.)589 4308 y(W)-8 b(e)23 b(need)e(the)g(Cotlar-Stein)g(lemma) f(that)i(w)m(e)g(recall)e(b)s(elo)m(w)h(\(see)h([T)-8 b(or])22 b(or)f([F)-8 b(ol1)q(]\):)448 4496 y Fz(Lemma)33 b(5.17)46 b Fo(L)-5 b(et)32 b Fl(f)p Ft(B)1320 4511 y FF(l)1347 4496 y Fl(g)1392 4511 y FF(l)q FG(2)p FF(L)1545 4496 y Fo(b)-5 b(e)32 b(a)g(family)h(of)f(op)-5 b(er)g(ators)35 b(in)d Ft(B)5 b Fy(\()p Fn(H)2819 4510 y Fq(1)2857 4496 y Ft(;)15 b Fn(H)2986 4510 y Fq(2)3026 4496 y Fy(\))32 b Fo(for)g(some)448 4608 y(Hilb)-5 b(ert)33 b(sp)-5 b(ac)g(es)34 b Fn(H)1110 4622 y Fq(1)1149 4608 y Fo(,)f Fn(H)1299 4622 y Fq(2)1337 4608 y Fo(.)42 b(Assume)33 b(that)822 4830 y Fy(sup)832 4909 y FF(l)q FG(2)p FF(L)987 4744 y Fm(X)974 4941 y FF(m)p FG(2)p FF(L)1147 4830 y Fy(max)1331 4729 y Fm(n)1392 4830 y Fl(k)p Ft(B)1506 4845 y FF(l)1532 4830 y Ft(B)1606 4793 y FG(\003)1601 4853 y FF(m)1668 4830 y Fl(k)1713 4782 y Fq(1)p FF(=)p Fq(2)1713 4863 y FF(B)s Fq(\()p Ff(H)1869 4872 y Fk(2)1905 4863 y Fq(\))1969 4830 y Ft(;)48 b Fl(k)p Ft(B)2161 4793 y FG(\003)2156 4853 y FF(l)2201 4830 y Ft(B)2270 4844 y FF(m)2336 4830 y Fl(k)2381 4782 y Fq(1)p FF(=)p Fq(2)2381 4863 y FF(B)s Fq(\()p Ff(H)2537 4872 y Fk(1)2573 4863 y Fq(\))2605 4729 y Fm(o)2691 4830 y Fy(=)25 b Ft(b)g(<)g Fl(1)p Ft(:)1897 5225 y Fy(58)p eop %%Page: 59 59 59 58 bop 448 573 a Fo(Then)49 b(the)g(sum)1077 505 y Fm(P)1173 600 y FF(l)q FG(2)p FF(L)1309 573 y Ft(B)1378 588 y FF(l)1459 573 y Fl(\021)54 b Ft(B)f Fo(exists)c(in)g(the)g(str)-5 b(ong)50 b(op)-5 b(er)g(ator)52 b(top)-5 b(olo)g(gy)51 b(and)448 686 y Fl(k)p Ft(B)5 b Fl(k)612 704 y FF(B)s Fq(\()p Ff(H)768 713 y Fk(1)804 704 y FF(;)p Ff(H)897 713 y Fk(2)932 704 y Fq(\))989 686 y Fl(\024)25 b Ft(b)p Fo(.)589 914 y Fy(Theorem)34 b(5.16)h(will)d(b)s(e)h(a)h(consequence)h (of)f(the)g(next)g(lemma.)51 b(W)-8 b(e)35 b(denote)f(b)m(y)448 1027 y Fl(k)21 b(\001)f(k)604 1041 y FF(u;v)737 1027 y Fy(the)30 b(norm)g(in)f Ft(B)5 b Fy(\()p Fn(H)1461 994 y FF(u)1506 1027 y Ft(;)15 b Fn(H)1662 994 y FF(v)1703 1027 y Fy(\).)448 1209 y Fz(Lemma)33 b(5.18)46 b Fo(L)-5 b(et)31 b Ft(L)25 b Fl(\032)g Fp(R)1448 1176 y FF(n)1531 1209 y Fo(such)30 b(that)h Fl(j)p Ft(l)17 b Fl(\000)e Ft(m)p Fl(j)24 b(\025)h Fy(const)p Ft(:)h(>)f Fy(0)31 b Fo(if)e Ft(l)r(;)15 b(m)30 b Fo(ar)-5 b(e)31 b(distinct)448 1321 y(p)-5 b(oints)35 b(of)e Ft(L)p Fo(,)f(and)i(let)f Ft(a)25 b(>)g Fy(2)p Ft(n)p Fo(.)43 b(Then)33 b(ther)-5 b(e)33 b(is)g Ft(C)f(>)25 b Fy(0)33 b Fo(such)g(that,)h(for)g(e)-5 b(ach)33 b(family)448 1434 y(of)j(op)-5 b(er)g(ators)39 b Ft(V)1009 1449 y FF(l)1066 1434 y Fl(2)30 b Ft(B)5 b Fy(\()p Fn(H)1381 1401 y FF(u)1426 1434 y Ft(;)15 b Fn(H)1582 1401 y FF(v)1623 1434 y Fy(\))36 b Fo(with)h Ft(V)1948 1449 y FF(l)2004 1434 y Fy(=)31 b(0)k Fo(for)i(al)5 b(l)36 b(but)f(a)h(\014nite)f(numb)-5 b(er)36 b(of)g Ft(l)r Fo(,)448 1547 y(the)d(fol)5 b(lowing)34 b(estimate)f(holds:)448 1762 y Fy(\(5.15\))784 1657 y Fm(\015)784 1712 y(\015)784 1766 y(\015)834 1693 y(P)930 1788 y FF(l)q FG(2)p FF(L)1051 1762 y Fy(e)1092 1724 y FG(\000)p FF(il)q(P)1251 1762 y Ft(V)1304 1777 y FF(l)1330 1762 y Fy(e)1370 1724 y FF(il)q(P)1475 1657 y Fm(\015)1475 1712 y(\015)1475 1766 y(\015)1526 1825 y FF(u;v)1652 1762 y Fl(\024)25 b Ft(C)37 b Fy(sup)1860 1840 y FF(l)q FG(2)p FF(L)2002 1762 y Fy(max)15 b Fl(f)q(kh)p Ft(Q)p Fl(i)2419 1724 y FF(a)2461 1762 y Ft(V)2514 1777 y FF(l)2540 1762 y Fl(k)2585 1776 y FF(u;v)2720 1762 y Ft(;)48 b Fl(k)p Ft(V)2891 1777 y FF(l)2917 1762 y Fl(h)p Ft(Q)p Fl(i)3059 1724 y FF(a)3101 1762 y Fl(k)3146 1776 y FF(u;v)3248 1762 y Fl(g)16 b Ft(:)448 2025 y Fz(Pro)s(of:)48 b Fy(Let)39 b(us)e(denote)i Ft(B)1443 2040 y FF(l)1508 2025 y Fl(\021)f Fy(e)1658 1992 y FG(\000)p FF(il)q(P)1817 2025 y Fl(h)p Ft(P)13 b Fl(i)1958 1992 y FF(v)2000 2025 y Ft(V)2053 2040 y FF(l)2079 2025 y Fl(h)p Ft(P)g Fl(i)2220 1992 y FG(\000)p FF(u)2320 2025 y Fy(e)2361 1992 y FF(il)q(P)2504 2025 y Fy(and)37 b(c)m(hec)m(k)j(that)f(the)g(h)m(y-)448 2138 y(p)s(otheses)e(of)h(the)f(Cotlar-Stein)g(lemma)f(are)i(satis\014ed.)61 b(F)-8 b(or)38 b(eac)m(h)g(couple)f Ft(l)r Fy(,)i Ft(m)e Fy(of)448 2251 y(p)s(oin)m(ts)30 b(of)g Ft(L)g Fy(w)m(e)h(estimate:)462 2447 y Fl(k)p Ft(B)576 2462 y FF(l)603 2447 y Ft(B)677 2410 y FG(\003)672 2470 y FF(m)738 2447 y Fl(k)26 b Fy(=)f Fl(kh)p Ft(P)13 b Fl(i)1091 2410 y FF(v)1133 2447 y Ft(V)1186 2462 y FF(l)1212 2447 y Fl(h)p Ft(P)g Fl(i)1353 2410 y FG(\000)p Fq(2)p FF(u)1489 2447 y Fy(e)1529 2410 y FF(i)p Fq(\()p FF(l)q FG(\000)p FF(m)p Fq(\))p FF(P)1806 2447 y Ft(V)1879 2410 y FG(\003)1859 2470 y FF(m)1926 2447 y Fl(h)p Ft(P)g Fl(i)2067 2410 y FF(v)2108 2447 y Fl(k)515 2585 y(\024)52 b(k)p Ft(V)736 2600 y FF(l)763 2585 y Fl(h)p Ft(Q)p Fl(i)905 2548 y FF(a)947 2585 y Fl(k)992 2599 y FF(u;v)1129 2585 y Fl(\001)21 b(kh)p Ft(P)13 b Fl(i)1361 2548 y FF(u)1407 2585 y Fl(h)p Ft(Q)p Fl(i)1549 2548 y FG(\000)p FF(a)1646 2585 y Fl(h)p Ft(P)g Fl(i)1787 2548 y FG(\000)p Fq(2)p FF(u)1923 2585 y Fy(e)1963 2548 y FF(i)p Fq(\()p FF(l)q FG(\000)p FF(m)p Fq(\))p FF(P)2240 2585 y Fl(h)p Ft(Q)p Fl(i)2382 2548 y FG(\000)p FF(a)2479 2585 y Fl(h)p Ft(P)g Fl(i)2620 2548 y FF(u)2666 2585 y Fl(k)36 b(\001)20 b(kh)p Ft(Q)p Fl(i)2979 2548 y FF(a)3022 2585 y Ft(V)3095 2548 y FG(\003)3075 2608 y FF(m)3141 2585 y Fl(k)3186 2599 y FG(\000)p FF(v)r(;)p FG(\000)p FF(u)3398 2585 y Ft(:)448 2782 y Fy(W)-8 b(e)43 b(ha)m(v)m(e)h Fl(k)p Ft(V)936 2796 y FF(m)1003 2782 y Fl(h)p Ft(Q)p Fl(i)1145 2749 y FF(a)1187 2782 y Fl(k)1232 2796 y FF(u;v)1378 2782 y Fy(=)h Fl(kh)p Ft(Q)p Fl(i)1681 2749 y FF(a)1723 2782 y Ft(V)1797 2749 y FG(\003)1776 2804 y FF(m)1843 2782 y Fl(k)1888 2796 y FG(\000)p FF(v)r(;)p FG(\000)p FF(u)2100 2782 y Fy(.)75 b(By)42 b(standard)f(comm)m(utator)j (esti-)448 2895 y(mates)32 b(\(see)g(Prop)s(osition)d(4.1.2)k(in)c ([ABG)q(]\))j(there)f(is)f(a)h(b)s(ounded)e(op)s(erator)i Ft(S)36 b Fy(suc)m(h)448 3008 y(that)i Fl(h)p Ft(P)13 b Fl(i)793 2975 y FF(u)838 3008 y Fl(h)p Ft(Q)p Fl(i)980 2975 y FG(\000)p FF(a)1077 3008 y Fl(h)p Ft(P)g Fl(i)1218 2975 y FG(\000)p Fq(2)p FF(u)1390 3008 y Fy(=)36 b Ft(S)5 b Fl(h)p Ft(P)13 b Fl(i)1699 2975 y FG(\000)p FF(u)1799 3008 y Fl(h)p Ft(Q)p Fl(i)1941 2975 y FG(\000)p FF(a)2039 3008 y Fy(,)38 b(th)m(us)e(the)h(middle)e(norm)h(in)f(the)i(last)448 3121 y(term)31 b(of)f(the)h(ab)s(o)m(v)m(e)g(inequalit)m(y)e(ma)m(y)i (b)s(e)f(ma)5 b(jorated)31 b(b)m(y)f Fl(k)p Ft(S)5 b Fl(k)31 b Fy(times)f(the)h(quan)m(tit)m(y:)488 3317 y Fl(kh)p Ft(P)13 b Fl(i)674 3280 y FG(\000)p FF(u)775 3317 y Fl(h)p Ft(Q)p Fl(i)917 3280 y FG(\000)p FF(a)1014 3317 y Fy(e)1055 3280 y FF(i)p Fq(\()p FF(l)q FG(\000)p FF(m)p Fq(\))p FF(P)1331 3317 y Fl(h)p Ft(Q)p Fl(i)1473 3280 y FG(\000)p FF(a)1570 3317 y Fy(e)1611 3280 y FG(\000)p FF(i)p Fq(\()p FF(l)q FG(\000)p FF(m)p Fq(\))p FF(P)1943 3317 y Fl(h)p Ft(P)g Fl(i)2084 3280 y FF(u)2129 3317 y Fl(k)26 b Fy(=)f Fl(kh)p Ft(Q)p Fl(i)2483 3280 y FG(\000)p FF(a)2580 3317 y Fl(h)p Ft(Q)c Fl(\000)f Fy(\()p Ft(m)g Fl(\000)g Ft(l)r Fy(\))p Fl(i)3124 3280 y FG(\000)p FF(a)3221 3317 y Fl(k)3266 3331 y FF(u;u)3372 3317 y Ft(:)448 3514 y Fy(Let)45 b(us)f(denote)g(b)m(y)h Ft(C)50 b Fy(a)45 b(generic)f(p)s(ositiv)m(e)f(\014nite)h(constan)m(t.)83 b(By)45 b(in)m(terp)s(olation)448 3627 y(b)s(et)m(w)m(een)31 b(0)g(and)f(an)g(in)m(teger)h Ft(N)k(>)25 b Fl(j)p Ft(u)p Fl(j)31 b Fy(the)f(ab)s(o)m(v)m(e)i(quan)m(tit)m(y)f(is)e(dominated)h (b)m(y)831 3846 y Ft(C)21 b Fl(kh)p Ft(Q)p Fl(i)1104 3808 y FG(\000)p FF(a)1202 3846 y Fl(h)p Ft(Q)f Fl(\000)g Fy(\()p Ft(m)h Fl(\000)f Ft(l)r Fy(\))p Fl(i)1746 3808 y FG(\000)p FF(a)1843 3846 y Fl(k)1888 3860 y FF(N)s(;N)2059 3846 y Fl(\024)25 b Ft(C)57 b Fy(sup)2257 3925 y FF(x)p FG(2)p Fc(R)2396 3906 y Fv(n)2482 3846 y Fy(sup)2450 3929 y FG(j)p FF(\013)p FG(j\024)p FF(N)2667 3741 y Fm(\014)2667 3796 y(\014)2667 3851 y(\014)2697 3846 y Ft(')2756 3798 y Fq(\()p FF(\013)p Fq(\))2756 3875 y FF(l)q(m)2861 3846 y Fy(\()p Ft(x)p Fy(\))2983 3741 y Fm(\014)2983 3796 y(\014)2983 3851 y(\014)3029 3846 y Ft(;)448 4119 y Fy(where)35 b Ft(')775 4134 y FF(l)q(m)864 4119 y Fy(\()p Ft(x)p Fy(\))e Fl(\021)g(h)p Ft(x)p Fl(i)1245 4086 y FG(\000)p FF(a)1342 4119 y Fl(h)p Ft(x)23 b Fl(\000)g Fy(\()p Ft(m)h Fl(\000)f Ft(l)r Fy(\))p Fl(i)1878 4086 y FG(\000)p FF(a)1975 4119 y Fy(.)54 b(F)-8 b(urthermore,)36 b(for)f(eac)m(h)h Ft(\013)g Fy(there)f(is)f(a)448 4232 y(constan)m(t)e Ft(c)854 4246 y FF(\013)934 4232 y Fy(suc)m(h)e(that)1178 4338 y Fm(\014)1178 4393 y(\014)1178 4447 y(\014)1209 4443 y Ft(')1268 4395 y Fq(\()p FF(\013)p Fq(\))1268 4472 y FF(l)q(m)1373 4443 y Fy(\()p Ft(x)p Fy(\))1495 4338 y Fm(\014)1495 4393 y(\014)1495 4447 y(\014)1551 4443 y Fl(\024)25 b Ft(c)1686 4457 y FF(\013)1766 4443 y Fl(j)p Ft(')1850 4458 y FF(l)q(m)1939 4443 y Fy(\()p Ft(x)p Fy(\))p Fl(j)h(\024)f Ft(C)d Fl(h)p Ft(l)g Fl(\000)e Ft(m)p Fl(i)2585 4405 y FG(\000)p FF(a)2682 4443 y Ft(:)448 4667 y Fy(Hence)39 b Fl(k)p Ft(B)840 4682 y FF(l)867 4667 y Ft(B)941 4634 y FG(\003)936 4689 y FF(m)1002 4667 y Fl(k)f(\024)g Ft(C)22 b Fl(h)p Ft(l)27 b Fl(\000)e Ft(m)p Fl(i)1581 4634 y FG(\000)p FF(a)1693 4667 y Fy(sup)1830 4689 y FF(l)q FG(2)p FF(L)1966 4667 y Fl(k)p Ft(V)2064 4682 y FF(l)2091 4667 y Fl(h)p Ft(Q)p Fl(i)2233 4634 y FF(a)2275 4667 y Fl(k)2320 4681 y FF(u;v)2422 4667 y Ft(:)38 b Fy(Similarly)c(w)m(e)k(obtain)g(the)448 4780 y(estimate)31 b Fl(k)p Ft(B)931 4747 y FG(\003)926 4807 y FF(l)971 4780 y Ft(B)1040 4794 y FF(m)1106 4780 y Fl(k)26 b(\024)f Ft(C)d Fl(h)p Ft(l)g Fl(\000)e Ft(m)p Fl(i)1650 4747 y FG(\000)p FF(a)1762 4780 y Fy(sup)1899 4802 y FF(l)q FG(2)p FF(L)2035 4780 y Fl(kh)p Ft(Q)p Fl(i)2222 4747 y FF(a)2265 4780 y Ft(V)2318 4795 y FF(l)2344 4780 y Fl(k)2389 4794 y FF(u;v)2521 4780 y Fy(whic)m(h)29 b(\014nally)f(yields)455 4976 y(max)16 b Fl(fk)p Ft(B)799 4991 y FF(l)826 4976 y Ft(B)900 4939 y FG(\003)895 4999 y FF(m)961 4976 y Fl(k)31 b Ft(;)46 b Fl(k)p Ft(B)1227 4939 y FG(\003)1222 4999 y FF(l)1266 4976 y Ft(B)1335 4990 y FF(m)1401 4976 y Fl(kg)27 b(\024)e Ft(C)c Fl(h)p Ft(l)i Fl(\000)d Ft(m)p Fl(i)1991 4939 y FG(\000)p FF(a)2102 4976 y Fy(max)c Fl(fkh)p Ft(Q)p Fl(i)2519 4939 y FF(a)2562 4976 y Ft(V)2615 4991 y FF(l)2641 4976 y Fl(k)2686 4990 y FF(u;v)2818 4976 y Ft(;)46 b Fl(k)p Ft(V)2987 4991 y FF(l)3013 4976 y Fl(h)p Ft(Q)p Fl(i)3155 4939 y FF(a)3197 4976 y Fl(k)3242 4990 y FF(u;v)3344 4976 y Fl(g)16 b Ft(:)1897 5225 y Fy(59)p eop %%Page: 60 60 60 59 bop 448 573 a Fy(The)28 b(h)m(yp)s(othesis)f Ft(a)e(>)g Fy(2)p Ft(n)j Fy(is)f(then)h(su\016cien)m(t)g(to)h(ensure)e(that)2635 505 y Fm(P)2731 600 y FF(m)p FG(2)p FF(L)2892 573 y Fl(h)p Ft(l)18 b Fl(\000)e Ft(m)p Fl(i)3174 540 y FG(\000)p FF(a=)p Fq(2)3366 573 y Fl(\024)448 686 y Fy(const)p Ft(:)46 b(<)e Fl(1)e Fy(indep)s(enden)m(tly)d(of)j Ft(l)k Fl(2)e Ft(L)p Fy(,)h(whic)m(h)c(means)h(that)g(the)g(h)m(yptheses)g(of) 448 799 y(Lemma)31 b(5.17)h(are)f(v)m(eri\014ed.)p 3371 791 67 67 v 589 1011 a(If)24 b Ft(H)750 1025 y Fq(0)813 1011 y Fy(is)f(as)h(in)e(the)i(statemen)m(t)i(of)e(Theorem)g(5.16)h 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FF(=)p Fq(2)448 1645 y Fy(and)44 b(the)g(fact)h(that)f(the)g(symmetric)f(op)s(erator)h Fl(h)p Ft(P)13 b Fl(i)2359 1613 y FF(s)2397 1645 y Fy(\()p Fl(j)p Ft(H)2533 1659 y Fq(0)2573 1645 y Fl(j)29 b Fy(+)g(1\))2807 1613 y FG(\000)p Fq(1)p FF(=)p Fq(2)3017 1645 y Fy(b)s(elongs)42 b(to)448 1758 y Fl(M)p Fy(\()p Fs(C)653 1772 y FF(L)706 1758 y Fy(\))f(\(b)s(ecause)g Ft(H)1240 1772 y Fq(0)1320 1758 y Fy(is)f(strictly)g(a\016liated)g(to)h Fs(C)2310 1772 y FF(L)2362 1758 y Fy(,)j(see)d(Prop)s(osition)e(3.16)k(and)448 1871 y Fl(x)p Fy(2.2\))33 b(reduce)d(the)g(problem)f(to)i(the)g(pro)s (of)e(of)i(the)f(statemen)m(t)448 2076 y(\(5.16\))483 b Fl(h)p Ft(P)13 b Fl(i)1302 2038 y FG(\000)p FF(s)p FG(\000)p FF(\025)1491 2076 y Ft(V)20 b Fl(h)p Ft(P)13 b Fl(i)1705 2038 y FG(\000)p FF(s)1822 2076 y Fl(2)25 b Fs(C)1969 2090 y FF(L)2082 2076 y Fy(for)30 b(some)61 b Ft(\025)26 b Fl(\025)f Fy(0)p Ft(:)448 2280 y Fy(These)40 b(general)h(remarks)f(allo)m(w)f(one)i(to)g(consider)e(the)i(case)g (when)f Ft(t)h Fy(=)h Ft(s)e Fy(in)f(the)448 2393 y(statemen)m(t)i(of)d (Theorem)h(5.16,)j(but)c(here)g(w)m(e)h(shall)e(consider)h(only)f(the)i (situation)448 2506 y Ft(t)47 b(<)g(s)p Fy(,)g(when)42 b(it)h(su\016ces)h(to)g(tak)m(e)h Ft(\025)i Fy(=)g(0)d(ab)s(o)m(v)m(e.) 81 b(W)-8 b(e)45 b(shall)d(pro)m(v)m(e)i(\(5.16\))i(b)m(y)448 2619 y(constructing)22 b(a)g(family)f(of)h(symmetric)f(op)s(erators)h Fl(f)p Ft(V)2319 2633 y FF(")2357 2619 y Fl(g)g Fy(of)g Fs(C)2580 2633 y FF(L)2654 2619 y Fy(whic)m(h)f(appro)m(ximates)448 2731 y Ft(V)51 b Fy(in)29 b(the)h(norm)g(of)h Ft(B)5 b Fy(\()p Fn(H)1380 2698 y FF(s)1417 2731 y Ft(;)15 b Fn(H)1573 2698 y FG(\000)p FF(s)1665 2731 y Fy(\).)589 2844 y(Cho)s(ose)44 b Ft(\022)50 b Fl(2)e Ft(C)1193 2811 y FG(1)1186 2867 y Fq(c)1267 2844 y Fy(\()p Fp(R)1362 2811 y FF(n)1415 2844 y Fy(\))d(with)d(0)49 b Fl(\024)e Ft(\022)k Fl(\024)c Fy(1)e(and)e Ft(\022)s Fy(\(0\))48 b(=)g(1,)g(and)43 b(set)i(\002)3282 2858 y FF(")3366 2844 y Fy(=)448 2957 y Ft(\022)s Fy(\()p Ft("Q)p 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FF(t)2745 4658 y Ft(;)15 b Fn(H)2901 4625 y FG(\000)p FF(t)2874 4680 y(a)2986 4658 y Fy(\).)39 b(By)25 b(in)m(ter-)448 4770 y(p)s(olation,)k(and)f(since)g Ft(t)d(<)g(s)p Fy(,)k(w)m(e)h(get)g(that)g(it)e(is)g(also)h(b)s(ounded) e(in)h Ft(B)5 b Fy(\()p Fn(H)3014 4737 y FF(t)2988 4793 y(\013)p FG(\000)p FF(a)3129 4770 y Ft(;)15 b Fn(H)3285 4737 y FG(\000)p FF(s)3258 4793 y(\013)3377 4770 y Fy(\).)448 4883 y(So)37 b(in)f(order)g(to)i(sho)m(w)f(that)g Fl(kh)p Ft(Q)p Fl(i)1678 4850 y FF(\013)1729 4883 y Fy(\()p Ft(V)1817 4898 y FF(l)q(;")1920 4883 y Fl(\000)24 b Ft(V)2068 4898 y FF(l)2094 4883 y Fy(\))p Fl(k)2174 4897 y FF(s;)p FG(\000)p FF(s)2355 4883 y Fl(!)37 b Fy(0)g(if)f Ft(")h Fl(!)f Fy(0)h(it)g(su\016ces)f(to)1897 5225 y(60)p eop %%Page: 61 61 61 60 bop 448 573 a Fy(pro)m(v)m(e)32 b(the)e(next)h(t)m(w)m(o)g (relations:)715 777 y Fl(kh)p Ft(P)13 b Fl(i)901 740 y FG(\000)p FF(s)994 777 y Fl(h)p Ft(Q)p Fl(i)1136 740 y FF(\013)1186 777 y Fy(\(\002)1292 791 y FF(")1349 777 y Fl(\000)20 b Fy(1\))p Fl(h)p Ft(Q)p Fl(i)1662 740 y FG(\000)p FF(a)1760 777 y Fl(h)p Ft(P)13 b Fl(i)1901 740 y FF(t)1931 777 y Fl(k)798 915 y Fy(=)83 b Fl(kh)p Ft(P)13 b Fl(i)1138 877 y FG(\000)p FF(s)1231 915 y Fl(h)p Ft(Q)p Fl(i)1373 877 y FF(\013)1423 915 y Fy(\(\002)1529 929 y FF(")1586 915 y Fl(\000)20 b Fy(1\))p Fl(h)p Ft(Q)p Fl(i)1899 877 y FG(\000)p FF(\013)2005 915 y Fl(h)p Ft(P)13 b Fl(i)2146 877 y FF(s)2203 915 y Fl(\001)21 b(h)p Ft(P)13 b Fl(i)2390 877 y FG(\000)p FF(s)2482 915 y Fl(h)p Ft(Q)p Fl(i)2624 877 y FF(\013)p FG(\000)p FF(a)2766 915 y Fl(h)p Ft(P)g Fl(i)2907 877 y FF(t)2938 915 y Fl(k)26 b(!)f Fy(0)715 1053 y Fl(kh)p Ft(P)13 b Fl(i)901 1015 y FG(\000)p FF(s)994 1053 y Fy(\(\002)1100 1067 y FF(")1157 1053 y Fl(\000)20 b Fy(1\))p Fl(h)p Ft(Q)p Fl(i)1470 1015 y FF(\013)p FG(\000)p FF(a)1613 1053 y Fl(h)p Ft(P)13 b Fl(i)1754 1015 y FF(t)1785 1053 y Fl(k)798 1191 y Fy(=)83 b Fl(kh)p Ft(P)13 b Fl(i)1138 1153 y FG(\000)p FF(s)1231 1191 y Fy(\(\002)1337 1205 y FF(")1394 1191 y 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b(example,)j(it)c(su\016ces)g(that)h Ft(V)66 b Fy(b)s(e)44 b(a)i(real)f(function)f(suc)m(h)h(that)448 725 y Fm(R)491 831 y Fc(R)543 812 y Fv(n)601 799 y Fl(j)p Ft(V)20 b Fy(\()p Ft(x)p Fy(\))p Fl(j)846 766 y Fq(2)887 799 y Fl(h)p Ft(x)p Fl(i)1009 766 y FG(\000)p FF(N)1147 799 y Fy(d)o Ft(x)34 b(<)g Fl(1)i Fy(for)g(some)g Ft(N)45 b Fy(and)35 b(that)i(there)f(is)f(an)g(op)s(en)g(subset)g(\012)448 912 y(of)h(the)f(unit)f(sphere)g Ft(S)1264 879 y FF(n)p FG(\000)p Fq(1)1435 912 y Fy(=)f Fl(f)p Ft(!)j Fl(2)d Fp(R)1831 879 y FF(n)1918 912 y Fl(j)g(j)p Ft(!)s Fl(j)h Fy(=)f(1)p Fl(g)p Fy(,)k(with)d Ft(S)2649 879 y FF(n)p FG(\000)p Fq(1)2810 912 y Fl(n)24 b Fy(\012)34 b(of)i(spherical)448 1024 y(measure)31 b(zero,)g(and)f(for)g(eac)m(h)h Ft(!)e Fl(2)24 b Fy(\012)30 b(there)h(is)e Ft(#)c(>)g Fy(0)31 b(suc)m(h)f(that:)1163 1156 y Fm(Z)1253 1182 y FG(1)1213 1362 y Fq(1)1343 1178 y Fm(\020)1430 1218 y Fy(1)p 1408 1258 91 4 v 1408 1342 a Ft(r)1452 1315 y FF(n)1523 1156 y Fm(Z)1574 1362 y FF(C)5 b Fq(\()p FF(!)r Fq(;)p FF(#;r)r Fq(\))1864 1279 y Fl(j)p Ft(V)21 b Fy(\()p Ft(x)p Fy(\))p Fl(j)2110 1242 y Fq(2)2165 1279 y Fy(d)p Ft(x)2268 1178 y Fm(\021)2332 1174 y Fk(1)p 2332 1186 31 3 v 2332 1227 a(2)2392 1279 y Fy(d)o Ft(r)28 b(<)d Fl(1)p Ft(:)448 1546 y Fy(Here)38 b Ft(C)7 b Fy(\()p Ft(!)s Fy(;)15 b Ft(#;)g(r)s Fy(\))38 b(is)f(the)g(truncated)h(cone)g(consisting)f(of)h (the)f(p)s(oin)m(ts)g Ft(x)g Fy(suc)m(h)g(that)448 1659 y Fl(h)p Ft(x;)15 b(!)s Fl(i)34 b Ft(>)e Fl(j)p Ft(x)p Fl(j)15 b Fy(cos)h Ft(#)35 b Fy(and)f Ft(r)h(<)d Fl(j)p Ft(x)p Fl(j)h Ft(<)f Fy(2)p Ft(r)s Fy(.)54 b(Unfortunately)-8 b(,)35 b(this)f(criterion)f(is)h(useless)448 1772 y(for)25 b(Klaus)e(p)s(oten)m(tials)h(if)g Ft(n)h Fy(=)g(1)g(\(in)e(this)h(case) i(one)f(can)g(in)e(fact)j(use)e([DeW)-8 b(o)r(])25 b(in)f(order)448 1885 y(to)33 b(sho)m(w)f(that)g(the)g(w)m(a)m(v)m(e)i(op)s(erators)e Fo(do)j(not)d Fy(exist)g(in)e(certain)i(cases\).)47 b(Ho)m(w)m(ev)m (er,)34 b(if)448 1998 y Ft(n)25 b Fl(\025)g Fy(2,)31 b(one)g(can)f(easily)g(get)h(non)m(trivial)e(results:)448 2185 y Fz(Prop)s(osition)37 b(5.20)46 b Fo(L)-5 b(et)30 b Ft(W)43 b Fo(b)-5 b(e)30 b(a)g(r)-5 b(e)g(al)32 b(squar)-5 b(e)30 b(inte)-5 b(gr)g(able)32 b(function)e(with)h(c)-5 b(omp)g(act)448 2298 y(supp)g(ort)48 b(on)e Fp(R)986 2265 y FF(n)1039 2298 y Fo(,)i Ft(n)h Fl(\025)f Fy(2)p Fo(,)h(and)e Ft(L)h Fl(\032)g Fp(R)1939 2265 y FF(n)2038 2298 y Fo(b)-5 b(e)45 b(a)h(sp)-5 b(arse)47 b(set.)81 b(L)-5 b(et)46 b Ft(H)52 b Fo(b)-5 b(e)46 b(a)f(self-)448 2411 y(adjoint)h(op)-5 b(er)g(ator)47 b(on)d Ft(L)1341 2378 y Fq(2)1381 2411 y Fy(\()p Fp(R)1476 2378 y FF(n)1529 2411 y Fy(\))g Fo(e)-5 b(qual)45 b(to)f Fy(\001)28 b(+)2174 2343 y Fm(P)2269 2438 y FF(l)q FG(2)p FF(L)2406 2411 y Ft(W)13 b Fy(\()p Ft(x)28 b Fl(\000)g Ft(l)r Fy(\))44 b Fo(on)h Fn(S)18 b Fy(\()p Fp(R)3161 2378 y FF(n)3214 2411 y Fy(\))44 b Fo(and)448 2524 y(let)36 b(us)g(denote)h(by)f Ft(H)1193 2538 y Fq(0)1267 2524 y Fo(the)h(usual)f(self-adjoint)h(r)-5 b(e)g(alization)38 b(of)f Fy(\001)e Fo(on)h Ft(L)2997 2491 y Fq(2)3037 2524 y Fy(\()p Fp(R)3132 2491 y FF(n)3185 2524 y Fy(\))p Fo(.)52 b(F)-7 b(or)448 2637 y Ft(!)28 b Fl(2)d Ft(S)680 2604 y FF(n)p FG(\000)p Fq(1)817 2637 y Ft(;)15 b(#)26 b(>)f Fy(0)p Ft(;)15 b(r)28 b(>)d Fy(0)p Fo(,)h(denote)f Ft(N)1737 2652 y FF(!)r(;#)1849 2637 y Fy(\()p Ft(r)s Fy(\))f Fo(the)g(numb)-5 b(er)25 b(of)f(p)-5 b(oints)25 b(in)f Ft(L)q Fl(\\)q Ft(C)7 b Fy(\()p Ft(!)s Fy(;)15 b Ft(#;)g(r)s Fy(\))p Fo(.)448 2750 y(Assume)33 b(that)i(ther)-5 b(e)34 b(is)f(an)h(op)-5 b(en)34 b(set)g Fy(\012)26 b Fl(\032)g Ft(S)2038 2717 y FF(n)p FG(\000)p Fq(1)2208 2750 y Fo(with)34 b Ft(S)2467 2717 y FF(n)p FG(\000)p Fq(1)2625 2750 y Fl(n)21 b Fy(\012)33 b Fo(of)g(me)-5 b(asur)g(e)35 b(zer)-5 b(o,)448 2863 y(and)40 b(for)f(e)-5 b(ach)39 b Ft(!)g Fl(2)d Fy(\012)i Fo(ther)-5 b(e)39 b(is)g Ft(#)d(>)f Fy(0)k Fo(such)g(that)2310 2790 y Fm(R)2370 2816 y FG(1)2353 2895 y Fq(1)2460 2788 y Fm(p)p 2551 2788 300 4 v 75 x Ft(N)2624 2878 y FF(!)r(;#)2736 2863 y Fy(\()p Ft(r)s Fy(\))p Ft(r)2894 2830 y FG(\000)p FF(n=)p Fq(2)3082 2863 y Fy(d)o Ft(r)g(<)c Fl(1)p Fo(.)448 2976 y(Then)e(the)g(wave)g(op)-5 b(er)g(ators)36 b(for)d(the)g(p)-5 b(air)34 b Fy(\()p Ft(H)r(;)15 b(H)2166 2990 y Fq(0)2206 2976 y Fy(\))33 b Fo(exist.)448 3163 y Fz(Pro)s(of:)48 b Fy(If)c Ft(K)57 b Fy(=)49 b(supp)14 b Ft(W)58 b Fy(and)44 b Ft(r)k Fy(is)c(large)h(enough,)k(then)c Ft(C)7 b Fy(\()p Ft(!)s Fy(;)15 b Ft(#;)g(r)s Fy(\))45 b(con)m(tains)448 3276 y(at)h(most)f Ft(c)p Fy(\()p Ft(n)p Fy(\))p Ft(N)1048 3291 y FF(!)r(;#)1161 3276 y Fy(\()p Ft(r)s Fy(\))g(sets)g(of)g(the)g (form)g Ft(l)32 b Fy(+)d Ft(K)52 b Fy(\()p Ft(l)g Fl(2)d Ft(L)p Fy(\))c(and)f(these)h(sets)h(are)448 3389 y(pairwise)21 b(disjoin)m(t.)36 b(Th)m(us,)24 b(if)d Ft(V)g Fy(\()p Ft(x)p Fy(\))k(=)1793 3321 y Fm(P)1889 3416 y FF(l)q FG(2)p FF(L)2026 3389 y Ft(W)13 b Fy(\()p Ft(x)5 b Fl(\000)g Ft(l)r Fy(\),)23 b(then)2604 3316 y Fm(R)2647 3421 y FF(C)5 b Fq(\()p FF(!)r Fq(;)p FF(#;r)r Fq(\))2938 3389 y 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b(for)c(the)g(p)-5 b(air)34 b Fy(\()p Ft(H)r(;)15 b(H)1597 4564 y Fq(0)1637 4550 y Fy(\))33 b Fo(exist.)448 4737 y Fy(F)-8 b(or)45 b(the)f(pro)s(of)g (it)f(su\016ces)h(to)h(note)g(that)g Ft(N)2099 4752 y FF(!)r(;#)2211 4737 y Fy(\()p Ft(r)s Fy(\))p Ft(\016)2365 4752 y FF(!)r(;#)2477 4737 y Fy(\()p Ft(r)s Fy(\))k Fl(\024)e Ft(c)2797 4751 y FF(n)2845 4737 y Ft(r)2889 4704 y FF(n)2979 4737 y Fy(for)d(large)g Ft(r)s Fy(.)448 4850 y(Observ)m(e)c(that)h(the) f(condition)e(of)i(the)g(corollary)f(is)g(satis\014ed)g(if)g(for)h(eac) m(h)h Ft(!)j Fl(2)c Fy(\012)448 4963 y(there)31 b(are)g Ft(#;)15 b(c;)g(")26 b(>)f Fy(0)31 b(suc)m(h)f(that)h Ft(\016)1688 4978 y FF(!)r(;#)1800 4963 y Fy(\()p Ft(r)s Fy(\))26 b Fl(\025)f Ft(cr)2119 4930 y Fq(2)p FF(=n)p Fq(+)p FF(")2354 4963 y Fy(for)30 b(all)f Ft(r)s Fy(.)1897 5225 y(62)p eop %%Page: 63 63 63 62 bop 448 573 a FA(6)135 b(App)t(endix)448 854 y Fz(6.1.)82 b(En)m(v)m(eloping)35 b Ft(C)1301 821 y FG(\003)1340 854 y Fz(-Algebras)448 1007 y Fy(The)44 b(purp)s(ose)f(of)h(this)f (paragraph)h(is)f(to)i(summarize)e(sev)m(eral)i(elemen)m(tary)f(facts) 448 1120 y(concerning)e(en)m(v)m(eloping)g Ft(C)1446 1087 y FG(\003)1485 1120 y Fy(-algebras)h(whic)m(h)e(w)m(e)i(need)f(in) f(the)i(main)e(text.)78 b(W)-8 b(e)448 1233 y(refer)28 b(to)h Fl(x)p Fy(2.7)g(in)e([Dix])h(for)g(details)f(in)g(the)h(case)h (when)e(an)h(appro)m(ximate)g(unit)e(exists)448 1346 y(\(whic)m(h)k(is)f(true)h(in)f(the)i(situation)e(of)i(crossed)f(pro)s (ducts\).)589 1459 y(Let)50 b(\()p Fl(A)p Ft(;)15 b Fl(k)34 b(\001)f(k)p Fy(\))50 b(b)s(e)e(a)i(Banac)m(h)g Fl(\003)p Fy(-algebra.)98 b(A)49 b Ft(C)2441 1426 y FG(\003)2480 1459 y Fy(-\(semi\)norm)g(on)g Fl(A)f Fy(is)h(a)448 1572 y(\(semi\)norm)30 b Fl(j)21 b(\001)f(j)31 b Fy(satisfying)e Fl(j)p Ft(S)5 b(T)13 b Fl(j)25 b(\024)h(j)p Ft(S)5 b Fl(j)20 b(\001)h(j)p Ft(T)13 b Fl(j)31 b Fy(and)e Fl(j)p Ft(S)5 b Fl(j)2396 1539 y Fq(2)2462 1572 y Fy(=)25 b Fl(j)p Ft(S)2644 1539 y FG(\003)2684 1572 y Ft(S)5 b Fl(j)30 b Fy(for)g(all)g Ft(S;)15 b(T)38 b Fl(2)25 b(A)p Fy(.)448 1684 y(W)-8 b(e)27 b(recall)f(that)g(eac)m(h)h(morphism)d (\(i.e.)i Fl(\003)p Fy(-morphism\))f(from)g Fl(A)h Fy(in)m(to)g(a)g Ft(C)2993 1651 y FG(\003)3032 1684 y Fy(-algebra)g(is)448 1797 y(a)h(con)m(traction.)40 b(Hence)27 b(w)m(e)f(ha)m(v)m(e)h Fl(j)11 b(\001)g(j)26 b(\024)f(k)11 b(\001)g(k)28 b Fy(for)d(eac)m(h)i Ft(C)2421 1764 y FG(\003)2460 1797 y Fy(-\(semi\)norm)f(on)f Fl(A)p Fy(.)39 b(Since)448 1910 y(the)26 b(upp)s(er)e(b)s(ound)g(of)i (an)m(y)g(family)e(of)i Ft(C)1842 1877 y FG(\003)1881 1910 y Fy(-\(semi\)norms)f(is)g(a)h Ft(C)2658 1877 y FG(\003)2697 1910 y Fy(-\(semi\)norm,)h(there)448 2023 y(is)36 b(a)g(largest)h Ft(C)999 1990 y FG(\003)1038 2023 y Fy(-\(semi\)norm)f(on)g Fl(A)14 b Fy(:)53 b(w)m(e)37 b(call)e(it)h Fo(the)i Ft(C)2494 1990 y FG(\003)2533 2023 y Fo(-\(semi\)norm)i(of)e Fl(A)e Fy(and)448 2136 y(w)m(e)30 b(denote)f(it)f(b)m(y)h Fl(k)17 b(\001)h(k)1236 2169 y(\003)1319 2136 y Fy(\(note)29 b(that)h Fl(k)17 b(\001)h(k)1900 2169 y(\003)1979 2136 y(\024)25 b(k)17 b(\001)h(k)p Fy(\).)41 b(The)28 b Ft(C)2583 2103 y FG(\003)2622 2136 y Fy(-algebra)h(obtained)f(b)m(y)448 2249 y(separation)i(and)f (completion)h(of)g(\()p Fl(A)p Ft(;)15 b Fl(k)20 b(\001)g(k)1935 2282 y(\003)1989 2249 y Fy(\))30 b(is)f(denoted)h Fl(A)2561 2282 y(\003)2645 2249 y Fy(\(with)f(norm)g(denoted)448 2362 y(again)i(b)m(y)f Fl(k)21 b(\001)f(k)973 2395 y(\003)1027 2362 y Fy(\),)31 b(and)f(is)f(called)h Fo(the)j(enveloping)g Ft(C)2313 2329 y FG(\003)2352 2362 y Fo(-algebr)-5 b(a)33 b(of)g Fl(A)p Fy(.)589 2475 y(Let)40 b Ft(\022)792 2489 y Fx(A)884 2475 y Fy(:)f Fl(A)g(!)g(A)1263 2508 y(\003)1355 2475 y Fy(b)s(e)f(the)h(canonical)f(morphism.)63 b(Then)38 b Ft(\022)2821 2489 y Fx(A)2913 2475 y Fy(is)f(con)m(tin)m(uous)448 2588 y(\(con)m(tractiv)m(e\))j(with)d(dense)f(range)i(and)f(it)g(is)f (injectiv)m(e)h(if)g(and)f(only)h(if)f Fl(k)26 b(\001)f(k)3202 2621 y(\003)3293 2588 y Fy(is)37 b(a)448 2701 y(norm)c(on)g Fl(A)p Fy(,)g(so)h(if)e(and)g(only)g(if)g(there)i(is)e(a)h Ft(C)2097 2668 y FG(\003)2136 2701 y Fy(-norm)g(on)g Fl(A)p Fy(.)48 b(In)33 b(this)f(case)i(w)m(e)f(sa)m(y)448 2814 y(that)h Fl(A)e Fy(is)g(an)h Ft(A)1044 2781 y FG(\003)1084 2814 y Fo(-algebr)-5 b(a)41 b Fy(and)32 b(w)m(e)i(iden)m(tify)d Fl(A)e(\032)h(A)2357 2847 y(\003)2410 2814 y Fy(,)k(th)m(us)e Fl(A)h Fy(b)s(ecomes)g(a)g(dense)448 2926 y Fl(\003)p Fy(-subalgebra)d(of)h Fl(A)1156 2960 y(\003)1240 2926 y Fy(and)e Fl(k)p Ft(A)p Fl(k)1574 2940 y FG(\003)1640 2926 y Fl(\024)c(k)p Ft(A)p Fl(k)31 b Fy(if)f Ft(A)25 b Fl(2)g(A)p Fy(.)589 3039 y(The)30 b(algebra)f Fl(A)1166 3072 y(\003)1249 3039 y Fy(ob)m(viously)g(has)g(the)h(follo)m(wing)e Fo(universal)k(pr)-5 b(op)g(erty)8 b Fy(:)44 b(if)28 b Fl(C)35 b Fy(is)29 b(a)448 3152 y Ft(C)520 3119 y FG(\003)559 3152 y Fy(-algebra)k(and)f Ft(\036)d Fy(:)f Fl(A)h(!)f(C)38 b Fy(is)31 b(a)i(morphism)d(then)i(there)h(is)e(a)i(unique)d(morphism) 448 3265 y Ft(\036)502 3298 y Fl(\003)590 3265 y Fy(:)35 b Fl(A)723 3298 y(\003)810 3265 y(!)f(C)41 b Fy(suc)m(h)35 b(that)h Ft(\036)f Fy(=)e Ft(\036)1683 3298 y Fl(\003)1761 3265 y(\016)24 b Ft(\022)1861 3279 y Fx(A)1915 3265 y Fy(.)56 b(As)36 b(a)g(consequence,)i(w)m(e)e(get)h(a)f(\(co)m(v)-5 b(ari-)448 3378 y(an)m(t\))30 b(functor)e(from)g(the)h(category)i(of)d (Banac)m(h)i Fl(\003)p Fy(-algebras)f(to)h(that)f(of)g Ft(C)3019 3345 y FG(\003)3058 3378 y Fy(-algebras.)448 3491 y(Indeed,)36 b(if)d Ft(\036)g Fy(:)g Fl(A)f(!)h(B)k Fy(is)d(a)h(morphism)e(b)s(et)m(w)m(een)i(t)m(w)m(o)i(Banac)m(h)e Fl(\003)p Fy(-algebras,)i(then)448 3604 y(clearly)30 b(there)h(is)e(a)i(morphism)d Ft(\036)1618 3637 y Fl(\003)1697 3604 y Fy(:)d Fl(A)1820 3637 y(\003)1899 3604 y(!)g(B)2075 3637 y(\003)2159 3604 y Fy(suc)m(h)30 b(that)h Ft(\036)2615 3637 y Fl(\003)2689 3604 y(\016)21 b Ft(\022)2786 3618 y Fx(A)2864 3604 y Fy(=)k Ft(\022)2991 3618 y Fx(B)3057 3604 y Fl(\016)c Ft(\036)3177 3637 y Fl(\003)3231 3604 y Fy(.)589 3717 y(It)31 b(is)e(easy)i(to)g(see)f(that)h Fo(if)h Ft(\036)h Fo(has)g(dense)g(r)-5 b(ange)33 b(then)g Ft(\036)2524 3750 y Fl(\003)2611 3717 y Fo(is)f(surje)-5 b(ctive)7 b Fy(.)40 b(But)31 b(w)m(e)448 3830 y(stress)g(that)g Ft(\036)949 3863 y Fl(\003)1035 3830 y Fo(c)-5 b(ould)34 b(b)-5 b(e)32 b(non-inje)-5 b(ctive)32 b(even)h(if)f Ft(\036)h Fo(is)f(inje)-5 b(ctive)7 b Fy(.)589 3943 y(F)-8 b(or)36 b(example,)f(if)f(there)h(are)f(sev)m(eral)h(distinct)e Ft(C)2331 3910 y FG(\003)2370 3943 y Fy(-norms)h(on)h Fl(A)f Fy(\(whic)m(h)f(is)h(the)448 4056 y(case)j(if)d Fl(A)f Fy(=)g Ft(L)1006 4023 y Fq(1)1046 4056 y Fy(\()p Ft(G)p Fy(\))j(and)f Ft(G)g Fy(is)g(a)h(non-amenable)e(lo)s(cally)g (compact)j(group\),)f(then)448 4168 y(there)j(is)e(a)h Ft(C)943 4135 y FG(\003)982 4168 y Fy(-norm)f Fl(j)26 b(\001)g(j)38 b Fy(on)g Fl(A)f Fy(distinct)g(from)g Fl(k)26 b(\001)f(k)2391 4202 y(\003)2446 4168 y Fy(.)63 b(So)38 b Fl(j)26 b(\001)f(j)39 b(\024)e(k)26 b(\001)g(k)3108 4202 y(\003)3200 4168 y Fy(and)37 b(if)448 4281 y Fl(A)521 4295 y FG(\017)591 4281 y Fy(is)31 b(the)g(algebra)g(obtained)f(b)m(y)h (completing)g(\()p Fl(A)p Ft(;)15 b Fl(j)21 b(\001)g(j)p Fy(\))32 b(then)e(there)i(is)e(a)h(canonical)448 4394 y(morphism)d Fl(A)946 4408 y FG(\017)1011 4394 y Fl(!)d(A)1200 4427 y(\003)1283 4394 y Fy(whic)m(h)30 b(is)f(surjectiv)m(e)h(but)g (not)h(injectiv)m(e.)589 4507 y(F)-8 b(or)31 b(similar)c(reasons)i(it)h (ma)m(y)g(happ)s(en)e(that)i(the)g(inclusion)c Fl(A)f Ft(,)-15 b Fl(!)25 b(B)32 b Fy(of)e(a)g(closed)448 4620 y Fl(\003)p Fy(-subalgebra)36 b Fl(A)g Fy(of)g(the)h(Banac)m(h)g Fl(\003)p Fy(-algebra)g Fl(B)i Fy(induces)34 b(a)j(morphism)c Fl(A)3109 4653 y(\003)3198 4620 y(!)i(B)3384 4653 y(\003)448 4733 y Fy(whic)m(h)28 b(is)g(not)h(injectiv)m(e.)40 b(So)29 b(if)f Fl(B)j Fy(is)d(an)h Ft(A)1934 4700 y FG(\003)1974 4733 y Fy(-algebra)g(then)g Fl(A)2600 4766 y(\003)2682 4733 y Fy(cannot)h(\(in)e(general\))448 4855 y(b)s(e)20 b(iden)m(ti\014ed)f(with)g(the)i(closure)p 1587 4782 73 4 v 20 w Fl(A)f Fy(of)h Fl(A)f Fy(in)g Fl(B)2024 4888 y(\003)2098 4855 y Fy(\(but)g(there)h(is)f(a)h(canonical)f(surjection) 448 4976 y Fl(A)521 5009 y(\003)600 4976 y(!)p 716 4903 V 25 w(A)p Fy(,)30 b(so)p 956 4903 V 31 w Fl(A)g Fy(is)f(a)i(quotien)m (t)f(of)h Fl(A)1761 5009 y(\003)1814 4976 y Fy(\).)1897 5225 y(63)p eop %%Page: 64 64 64 63 bop 589 573 a Fy(Suc)m(h)26 b(unpleasan)m(t)g(features)g(do)h (not)f(o)s(ccur,)h(ho)m(w)m(ev)m(er,)i(in)c(the)i(case)g(of)g(ideals)e (\(b)m(y)448 686 y Fo(ide)-5 b(al)42 b Fy(in)29 b(a)i(Banac)m(h)h Fl(\003)p Fy(-algebra)g(w)m(e)f(mean)g(\\closed)g Fl(\003)p Fy(-ideal"\),)g(as)g(a)g(consequence)h(of)448 799 y(the)f(follo)m(wing) e(result.)448 978 y Fz(Theorem)34 b(6.1)46 b Fo(L)-5 b(et)1107 1211 y Fy(0)46 b Fl(\000)-42 b(\000)-20 b(\000)f(\000)-42 b(!)46 b(J)1716 1150 y Ft(\036)1619 1211 y Fl(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)46 b(A)2127 1150 y Ft( )2033 1211 y Fl(\000)-42 b(\000)-20 b(\000)f(\000)-42 b(!)46 b(B)j(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)46 b Fy(0)448 1366 y Fo(b)-5 b(e)33 b(an)g(exact)g(se)-5 b(quenc)g(e)32 b(of)h(Banach)h Fl(\003)p Fo(-algebr)-5 b(as.)43 b(Then)1036 1622 y Fy(0)k Fl(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)46 b(J)1485 1655 y(\003)1656 1545 y Ft(\036)1710 1578 y Fl(\003)1585 1622 y(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)46 b(A)1953 1655 y(\003)2121 1545 y Ft( )2180 1578 y Fl(\003)2053 1622 y(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)47 b(B)2409 1655 y(\003)2509 1622 y(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)46 b Fy(0)448 1808 y Fo(is)33 b(an)g(exact)g(se) -5 b(quenc)g(e)32 b(of)h Ft(C)1459 1775 y FG(\003)1498 1808 y Fo(-algebr)-5 b(as.)448 1987 y Fz(Pro)s(of:)48 b Fy(a\))21 b(W)-8 b(e)21 b(need)f(only)g(one)g(non-trivial)e(result,)k (namely)d(the)i(Theorem)f(VI.19.11)448 2100 y(from)31 b([F)-8 b(eD)q(],)32 b(whic)m(h)d(sa)m(ys)i(that)h(if)e Fl(I)36 b Fy(is)30 b(an)h(ideal)e(in)h(a)h(Banac)m(h)h Fl(\003)p Fy(-algebra)f Fl(A)g Fy(and)f Fr(\031)448 2213 y Fy(is)i(a)g(non-degenerate)i(represen)m(tation)e(of)h Fl(I)38 b Fy(on)32 b(a)h(Hilb)s(ert)d(space)j Fn(H)27 b Fy(,)33 b(the)f(there)h(is)448 2326 y(a)h(unique)d(represen)m(tation) 40 b Fz(~)-58 b Fr(\031)36 b Fy(of)e Fl(A)f Fy(on)g Fn(H)60 b Fy(whic)m(h)32 b(extends)h Fr(\031)s Fy(.)49 b(This)32 b(implies)e(that)448 2439 y(the)j Ft(C)679 2406 y FG(\003)718 2439 y Fy(-seminorm)e Fl(k)23 b(\001)e(k)1324 2406 y FG(I)1324 2463 y Fl(\003)1407 2439 y Fy(of)33 b Fl(I)38 b Fy(is)32 b(equal)g(to)h(the)g(restriction)f(of)h(the)f Ft(C)2983 2406 y FG(\003)3022 2439 y Fy(-seminorm)448 2552 y Fl(k)21 b(\001)f(k)604 2519 y FG(A)604 2576 y Fl(\003)696 2552 y Fy(of)30 b Fl(A)g Fy(to)h Fl(I)7 b Fy(.)40 b(Indeed,)30 b(note)h(that)g(e.g.)h(the)e Ft(C)2250 2519 y FG(\003)2289 2552 y Fy(-norm)g(of)g Fl(A)p Fy(,)h(is)e(also)h (giv)m(en)h(b)m(y:)590 2746 y Fl(k)p Ft(S)5 b Fl(k)741 2709 y FG(A)741 2769 y Fl(\003)885 2746 y Fy(=)83 b(sup)o Fl(fk)p Fr(\031)t Fy(\()p Ft(S)5 b Fy(\))p Fl(k)27 b(j)e Fr(\031)34 b Fy(is)29 b(a)i(represen)m(tation)f(of)h Fl(Ag)885 2884 y Fy(=)83 b(sup)o Fl(fk)p Fr(\031)t Fy(\()p Ft(S)5 b Fy(\))p Fl(k)27 b(j)e Fr(\031)34 b Fy(is)29 b(a)i(non-degenerate)h(represen)m(tation)e(of)h Fl(Ag)448 3078 y Fy(b\))36 b(W)-8 b(e)37 b(apply)d(these)i(remarks)f(with)f(the)i (c)m(hoice)h Fl(I)j Fy(=)33 b Ft(\036)p Fy(\()p Fl(J)17 b Fy(\))35 b(=)e(k)m(er)16 b Ft( )s Fy(.)57 b(Since)34 b Ft(\036)i Fy(is)448 3191 y(a)k(bijectiv)m(e)f(morphism)e(of)i Fl(J)56 b Fy(on)m(to)40 b Fl(I)7 b Fy(,)41 b(w)m(e)e(ha)m(v)m(e)i Fl(k)p Ft(\036)p Fy(\()p Ft(S)5 b Fy(\))p Fl(k)2539 3158 y FG(I)2539 3215 y Fl(\003)2630 3191 y Fy(=)40 b Fl(k)p Ft(S)5 b Fl(k)2892 3153 y FG(J)2892 3216 y Fl(\003)2957 3191 y Fy(,)42 b(hence)d(also)448 3304 y Fl(k)p Ft(S)5 b Fl(k)599 3266 y FG(J)599 3329 y Fl(\003)700 3304 y Fy(=)35 b Fl(k)p Ft(\036)p Fy(\()p Ft(S)5 b Fy(\))p Fl(k)1081 3271 y FG(A)1081 3328 y Fl(\003)1144 3304 y Fy(.)59 b(So)36 b Ft(\036)h Fy(is)f(an)g(isometry)g(of)h(\()p Fl(J)17 b Ft(;)e Fl(k)25 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y(of)26 b Fl(B)j Fy(and)c(the)h(represen)m(tations)g(of)g Fl(A)g Fy(whic)m(h)e(are)j(zero)g(on)e Fl(I)33 b Fy(\(denote)26 b(b)m(y)g(Rep)15 b(\()p Fl(A)p Ft(;)g Fl(I)7 b Fy(\))448 3990 y(the)31 b(set)g(of)f(unitary)f(equiv)-5 b(alence)30 b(classes)g(of)h(suc)m(h)f(represen)m(tations\):)774 4184 y Fl(k)p Ft( )s Fy(\()p Ft(S)5 b Fy(\))p Fl(k)1057 4147 y FG(B)1057 4207 y Fl(\003)1194 4184 y Fy(=)83 b(sup)o Fl(fk)p Fr(\031)24 b Fl(\016)d Ft( )s Fy(\()p Ft(S)5 b Fy(\))p Fl(k)27 b(j)e Fr(\031)34 b Fy(is)c(a)g(represen)m(tation)h (of)f Fl(B)s(g)1194 4322 y Fy(=)83 b(sup)o Fl(fk)p Fr(\032)p Fy(\()p Ft(S)5 b Fy(\))p Fl(k)27 b(j)e Fr(\032)g Fl(2)g Fy(Rep)15 b(\()p Fl(A)p Ft(;)g Fl(I)7 b Fy(\))p Fl(g)448 4516 y Fy(The)29 b(map)h Fr(\032)24 b Fl(7!)i Fr(\032)1088 4557 y Fl(\003)1171 4516 y Fy(is)j(a)h(bijectiv)m(e)f(corresp)s (ondence)g(b)s(et)m(w)m(een)h(the)g(represen)m(tations)448 4629 y(of)k Fl(A)e Fy(and)h(the)g(represen)m(tations)g(of)g Fl(A)1811 4662 y(\003)1897 4629 y 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