Content-Type: multipart/mixed; boundary="-------------0108070316380" This is a multi-part message in MIME format. ---------------0108070316380 Content-Type: text/plain; name="01-298.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-298.keywords" compactification, invariant decompositions, Schrodinger operator, essential spectrum, anisotropic potential, crossed product ---------------0108070316380 Content-Type: application/postscript; name="ess.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ess.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Title: Ess.dvi %%Pages: 19 1 %%BoundingBox: 0 0 596 843 %%EndComments %%BeginProcSet: tex.pro /TeXDict 200 dict def TeXDict begin /N /def load def /B{bind def}N /S /exch load def /X{S N}B /TR /translate load N /isls false N /vsize 10 N /@rigin{ isls{[0 1 -1 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale Resolution VResolution vsize neg 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Fy(and)g Fu(b)f Fy(are)g(t)n(w)n(o)g(con)n(tin)n(uous)g(functions)h(on)418 3986 y Fu(S)469 3998 y Fq(X)532 3986 y Fy(,)34 b(the)g(unit)f(sphere)g(in)g Fu(X)7 b Fy(.)52 b(Then)33 b(the)g(essen)n(tial)f(sp)r(ectrum)h(of)g Fu(H)39 b Fy(=)31 b Fu(H)2883 3998 y Fs(0)2943 3986 y Fy(+)21 b Fu(V)52 b Fy(is)33 b(giv)n(en)418 4086 y(b)n(y)24 b Fu(\033)-51 b(\033)524 4082 y(\033)575 4098 y Fs(ess)661 4086 y Fy(\()p Fu(H)7 b Fy(\))23 b(=)g Ft([)967 4098 y Fq(\022)r Fn(2)p Fq(S)1087 4106 y Fm(X)1142 4086 y Fu(\033)-51 b(\033)1136 4082 y(\033)1186 4086 y Fy(\()p Fu(H)1287 4098 y Fq(\022)1325 4086 y Fy(\),)27 b(where)d Fu(\033)-51 b(\033)1638 4082 y(\033)1688 4086 y Fy(\()p Fu(H)1789 4098 y Fq(\022)1827 4086 y Fy(\))27 b(is)g(the)g(\(band\))g(sp)r (ectrum)g(of)g(the)g(Sc)n(hr\177)-42 b(odinger)418 4186 y(op)r(erator)26 b Fu(H)822 4198 y Fq(\022)883 4186 y Fy(:=)c Fu(H)1062 4198 y Fs(0)1118 4186 y Fy(+)c Fu(V)1249 4198 y Fq(\022)1315 4186 y Fy(with)28 b(p)r(erio)r(dic)f(p)r(oten)n(tial)h Fu(V)2221 4198 y Fq(\022)2259 4186 y Fy(\()p Fu(x)p Fy(\))c(=)f Fu(a)p Fy(\()p Fu(\022)r Fy(\))p Fu(W)12 b Fy([)p Fu(x)19 b Fy(+)f Fu(b)p Fy(\()p Fu(\022)r Fy(\)].)518 4285 y FA(Example)46 b(0.2.)73 b Fy(Set)41 b Fu(X)50 b Fy(:=)44 b Fr(R)1610 4255 y Fs(2)1697 4285 y Ft(\021)g Fr(R)1860 4297 y Fs(1)1930 4285 y Ft(\002)26 b Fr(R)2075 4297 y Fs(2)2158 4285 y Fy(and)40 b Fu(V)19 b Fy(\()p Fu(x)2478 4297 y Fs(1)2516 4285 y Fu(;)14 b(x)2600 4297 y Fs(2)2638 4285 y Fy(\))44 b(:=)g Fu(V)2894 4297 y Fs(1)2931 4285 y Fy(\()p Fu(x)3010 4297 y Fs(1)3048 4285 y Fy(\))p Fu(V)3128 4297 y Fs(2)3166 4285 y Fy(\()p Fu(x)3245 4297 y Fs(2)3283 4285 y Fy(\))27 b(+)418 4386 y Fu(V)466 4398 y Fs(0)504 4386 y Fy(\()p Fu(x)583 4398 y Fs(1)621 4386 y Fu(;)14 b(x)705 4398 y Fs(2)742 4386 y Fy(\))25 b(where)g Fu(V)1085 4398 y Fs(0)1147 4386 y Fy(deca)n(ys)f(at)g(in\014nit)n(y)h(and)g Fu(V)1996 4398 y Fq(j)2031 4386 y Fy(\()p Fu(x)2110 4398 y Fq(j)2146 4386 y Fy(\))e Ft(!)g Fu(c)2343 4350 y Fn(\006)2343 4409 y Fq(j)2423 4386 y Ft(2)g Fr(R)31 b Fy(when)25 b Fu(x)2847 4398 y Fq(j)2905 4386 y Ft(!)e(\0061)p Fy(.)36 b(Then)416 4500 y Fu(\033)-51 b(\033)410 4496 y(\033)460 4512 y Fs(ess)547 4500 y Fy(\()p Fu(H)7 b Fy(\))28 b(=)e Fu(\033)-51 b(\033)800 4496 y(\033)850 4500 y Fy(\()p Fu(H)958 4464 y Fn(\000)951 4522 y Fs(1)1014 4500 y Fy(\))21 b Ft([)e Fu(\033)-51 b(\033)1134 4496 y(\033)1185 4500 y Fy(\()p Fu(H)1293 4464 y Fs(+)1286 4522 y(1)1348 4500 y Fy(\))21 b Ft([)e Fu(\033)-51 b(\033)1468 4496 y(\033)1519 4500 y Fy(\()p Fu(H)1627 4464 y Fn(\000)1620 4522 y Fs(2)1683 4500 y Fy(\))21 b Ft([)e Fu(\033)-51 b(\033)1803 4496 y(\033)1854 4500 y Fy(\()p Fu(H)1962 4464 y Fs(+)1955 4522 y(2)2018 4500 y Fy(\).)46 b(W)-7 b(e)32 b(set)e Fu(H)2474 4464 y Fn(\006)2467 4523 y Fq(j)2559 4500 y Fy(:=)e Fu(H)2744 4512 y Fs(0)2801 4500 y Fy(+)21 b Fu(V)2953 4464 y Fn(\006)2935 4523 y Fq(j)3040 4500 y Fy(where,)32 b(for)418 4614 y(example,)27 b Fu(V)833 4579 y Fs(+)814 4636 y(1)888 4614 y Fy(\()p Fu(x)967 4626 y Fs(1)1005 4614 y Fu(;)14 b(x)1089 4626 y Fs(2)1127 4614 y Fy(\))23 b(:=)g Fu(c)1329 4579 y Fs(+)1329 4636 y(2)1384 4614 y Fu(V)1432 4626 y Fs(1)1470 4614 y Fy(\()p Fu(x)1549 4626 y Fs(1)1587 4614 y Fy(\).)518 4714 y(In)34 b(b)r(oth)g(examples)f(the)h(essen)n(tial)f (sp)r(ectrum)h(of)g Fu(H)41 b Fy(is)33 b(\(the)i(closure)d(of)6 b(\))35 b(the)f(union)g(of)418 4813 y(the)h(usual)f(sp)r(ectra)g(of)g(a)g (family)h(of)f(simpler)g(Sc)n(hr\177)-42 b(odinger)33 b(op)r(erators)f(whic)n (h)j(w)n(e)f(decided)418 4913 y(to)f(call)f Fv(asymptotic)k(Hamiltonians)p Fy(.)54 b(In)33 b(fact,)i(the)e(same)g(is)g(true)f(for)h(the)g(HWZ)g(theorem) 418 5013 y(for)24 b(whic)n(h)f("the)h(asymptotic)g(Hamiltonians")f(are)g (usually)g(called)h("subhamiltonians")e(and)418 5112 y(are)30 b(obtained)g(from)g(the)h(total)g(N-b)r(o)r(dy)f(Hamiltonian)h(b)n(y)f(thro)n (wing)g(a)n(w)n(a)n(y)e(some)i(suitable)p eop %%Page: 3 3 bop 1026 314 a Fw(C*{ALGEBRAS)29 b(AND)f(THE)g(ESSENTIAL)g(SPECTR)n(UM)601 b(3)418 513 y Fy(in)n(terparticle)30 b(p)r(oten)n(tials.)45 b(The)31 b(reason)e(for)h(our)g(c)n(hoice)f(of)i(terminology)e(\(whic)n(h)i (also)e(can)418 613 y(b)r(e)f(justi\014ed)g(for)e(the)i(N-b)r(o)r(dy)f (case\))g(is)g(the)g(follo)n(wing:)36 b(In)27 b(b)r(oth)h(situations)f(the)g (p)r(oten)n(tials)418 712 y(can)g(b)r(e)g(extended)h(con)n(tin)n(uously)e(to) h(v)n(ery)f(simple)h(compacti\014cations)f(of)h(the)g(space)g Fu(X)33 b Fy(and)418 812 y(the)c(asymptotic)e(Hamiltonians)h(are)f (constructed)h(with)g(restrictions)f(of)h(this)g(extension)g(to)418 911 y(suitable)e(pieces)f(of)h(the)h(b)r(oundary)e Fu(Z)6 b Fy(.)36 b(In)26 b(fact)g(our)f(main)h(con)n(tribution)f(will)h(b)r(e)h(to)e (explain)418 1015 y(what)i("suitable")f(means.)36 b(W)-7 b(e)28 b(describ)r(e)e(the)i(compacti\014cations)2587 994 y(^)2563 1015 y Fu(X)i Fy(=)22 b Fu(X)i Ft(t)17 b Fu(Z)33 b Fy(for)27 b(our)f(t)n(w)n(o)418 1114 y(cases)33 b(brie\015y;)38 b(a)c(thorough)f (discussion)h(in)h(m)n(uc)n(h)f(more)f(general)g(circumstances)h(ma)n(y)g(b)r (e)418 1214 y(found)28 b(in)g(the)g(main)f(b)r(o)r(dy)h(of)g(the)g(article.) 518 1314 y(In)39 b(Example)f(0.1.)70 b(one)39 b(can)g(tak)n(e)f Fu(Z)48 b Fy(:=)42 b Fu(S)1994 1326 y Fq(X)2083 1314 y Ft(\002)25 b Fr(T)2229 1284 y Fs(2)2266 1314 y Fy(,)42 b(where)c Fr(T)2638 1284 y Fs(2)2717 1314 y Fy(:=)k Fu(X=)p Fr(Z)3018 1284 y Fs(2)3088 1314 y Fy(is)d(the)h(2-)418 1413 y(dimensional)23 b(torus)f(\(a)h Fr(Z)1238 1383 y Fs(2)1269 1413 y Fy(-p)r(erio)r(dic)g(function)g(on)g Fu(X)30 b Fy(is)23 b(iden)n(ti\014ed)h(with)f(a)g(function)h(on)f Fr(T)3315 1383 y Fs(2)3352 1413 y Fy(\).)418 1513 y(But)36 b Fu(Z)42 b Fy(ma)n(y)36 b(also)f(b)r(e)h(represen)n(ted)f(as)g(the)i (disjoin)n(t)f(union)g Fu(Z)43 b Fy(=)37 b Ft(t)2687 1525 y Fq(\022)r Fn(2)p Fq(S)2807 1533 y Fm(X)2864 1513 y Fr(T)2920 1483 y Fs(2)2920 1537 y Fq(\022)2993 1513 y Fy(with)f(all)g Fr(T)3370 1483 y Fs(2)3370 1537 y Fq(\022)418 1613 y Fy(equal)25 b(with)i Fr(T)879 1582 y Fs(2)941 1613 y Fy(and)f(the)g(restrictions)f Fu(V)1719 1625 y Fq(\022)1783 1613 y Fy(of)h Fu(V)45 b Fy(to)25 b(the)i(\014b)r(ers)e Fr(T)2488 1582 y Fs(2)2488 1636 y Fq(\022)2551 1613 y Fy(in)n(terpreted)g(as)h(functions)418 1712 y(on)h Fu(X)34 b Fy(giv)n(e)27 b(the)h(asymptotic)f(Hamiltonians.)518 1816 y(In)37 b(Example)g(0.2.)64 b(the)38 b(compacti\014cation)1989 1795 y(^)1965 1816 y Fu(X)44 b Fy(is)37 b(the)h("square")p 2678 1748 98 4 v 35 w Fr(R)2732 1828 y Fs(1)2800 1816 y Ft(\002)p 2889 1748 V 24 w Fr(R)2943 1828 y Fs(2)2986 1816 y Fy(.)66 b(By)p 3215 1748 60 4 v 37 w Fr(R)43 b Fy(w)n(e)418 1915 y(denote)28 b(the)g(t)n(w)n(o-p)r(oin)n(t)e(\(radial\))h(compacti\014cation)g Fr(R)d Ft(t)19 b(f\0001g)e(t)i(f)p Fy(+)p Ft(1g)p Fy(.)35 b(W)-7 b(e)28 b(obtain)g(the)418 2023 y(four)36 b(asymptotic)h(Hamiltonians)f (\014rst)g(b)n(y)g(extending)h Fu(V)56 b Fy(to)2488 2002 y(^)2464 2023 y Fu(X)43 b Fy(and)37 b(then)g(restricting)f(it)418 2123 y(resp)r(ectiv)n(ely)31 b(to)h(the)g(four)f(closed)g(edges,)h(eac)n(h)f(one)h (iden)n(tical)f(to)h(a)f(cop)n(y)g(of)p 2966 2055 V 32 w Fr(R)6 b Fy(.)49 b(No)n(w)32 b(the)418 2222 y(decomp)r(osition)27 b(of)h Fu(Z)33 b Fy(is)27 b(no)h(longer)e(a)h(partition,)g(as)g(in)h(the)g (\014rst)g(case.)518 2322 y(Ob)n(viously)-7 b(,)26 b(there)g(are)h(man)n(y)f (other)g(w)n(a)n(ys)g(of)h(decomp)r(osing)f(the)i(b)r(ounderies)e(and)h (these)418 2422 y(w)n(ould)d(ha)n(v)n(e)e(led)i(to)g(wrong)f(results.)35 b(What)24 b(mak)n(es)f(the)h(ab)r(o)n(v)n(e)f(decomp)r(ositions)g("correct"?) 418 2521 y(The)32 b(formalism)f(w)n(e)h(shall)g(dev)n(elop)f(sho)n(ws)g(that) h(this)g(is)g(connected)g(to)g(the)g(fact)g(that)h(the)418 2621 y(kinetic)e(energy)f Fu(H)1029 2633 y Fs(0)1095 2621 y Fy(=)e Fu(h)p Fy(\()p Fu(P)12 b Fy(\))31 b(m)n(ust)g(also)f("liv)n(e")f(on)i (eac)n(h)f(mem)n(b)r(er)g(of)h(the)h(decomp)r(osition)418 2721 y(and)27 b(this)g(forces)e(it)i(to)g(b)r(e)g(closed)f(and)g(in)n(v)-5 b(arian)n(t)26 b(under)g(translations.)36 b(If)27 b(w)n(e)f(further)h(ask)e (a)418 2820 y(natural)c(minimalit)n(y)h(condition,)h(the)f(expression)f(of)g (the)i(essen)n(tial)d(sp)r(ectrum)i(is)g(completely)418 2920 y(determined.)46 b(In)31 b(particular,)f(eac)n(h)g(piece)h(giving)f(an)g (asymptotic)g(Hamiltonian)g(will)h(b)r(e)g(a)418 3020 y(quasi-orbit)26 b(of)i(the)g(dynamical)f(system)g(de\014ned)h(on)f(the)h(b)r(oundary)-7 b(.)518 3120 y(Let)35 b(us)g(sa)n(y)f(some)g(w)n(ords)g(on)g(the)i(lev)n(el)e (of)h(generalit)n(y)f(in)h(whic)n(h)g(w)n(e)f(place)h(ourselv)n(es.)418 3219 y(Un)n(til)j(no)n(w)f(in)h(this)g(in)n(tro)r(ductiv)n(e)f(discussion)f (the)i(space)f Fu(X)44 b Fy(has)37 b(b)r(een)h Fr(R)2892 3189 y Fq(n)2943 3219 y Fy(,)i(for)d(reasons)418 3319 y(of)h(simplicit)n(y)-7 b(.)67 b(W)-7 b(e)38 b(feel)g(that)g(the)g(natural)f(con)n(text)g(is)h(to)f (tak)n(e)g Fu(X)44 b Fy(a)37 b(lo)r(cally)g(compact,)418 3418 y(ab)r(elian)g(group)f(\(see)h(also)f([6]\);)42 b(this)37 b(w)n(orks)f(w)n (ell)g(for)h(almost)g(all)f(our)h(assertions.)63 b(Only)418 3518 y(a)32 b(non-compact)g Fu(X)39 b Fy(is)33 b(relev)-5 b(an)n(t.)51 b(The)33 b(particular)e(case)h Fu(X)38 b Fy(=)31 b Fr(Z)2558 3488 y Fq(n)2630 3518 y Fy(will)i(b)r(e)g(imp)r(ortan)n(t)g(for)418 3618 y(p)r(eople)f(in)n(terested)g(in)g(\014nite-di\013erence)h(op)r (erators.)48 b(Man)n(y)31 b(results)h(extend)g(easily)g(to)g(the)418 3717 y(non-ab)r(elian)h(case.)55 b(Because)32 b(our)h(lac)n(k)g(of)h (familiarit)n(y)f(with)h(this)g(situation,)h(w)n(e)f(decided)418 3817 y(ho)n(w)n(ev)n(er)25 b(to)h(tak)n(e)g Fu(X)33 b Fy(ab)r(elian.)j(W)-7 b(e)27 b(hop)r(e)g(to)f(b)r(e)h(able)f(to)h(treat)f(more)f(general)h (situations)g(in)418 3917 y(the)20 b(future.)35 b(The)20 b("free)f (Hamiltonian")g Fu(H)1766 3929 y Fs(0)1823 3917 y Fy(will)h(ha)n(v)n(e)f(the) h(form)f Fu(h)p Fy(\()p Fu(P)12 b Fy(\))20 b(for)g(some)f(con)n(tin)n(uous,) 418 4016 y(prop)r(er,)28 b(real)f(function)i Fu(h)g Fy(de\014ned)g(on)f(the)g (dual)h(group)e(of)h Fu(X)7 b Fy(.)39 b(W)-7 b(e)29 b(notice)f(that)h(the)g (c)n(hoice)418 4116 y Fu(h)p Fy(\()p Fu(p)p Fy(\))f(=)p Ft(j)g Fu(p)g Ft(j)809 4086 y Fs(2)877 4116 y Fy(do)r(esn't)j(alw)n(a)n(ys)e(mak)n (e)h(sense.)45 b(And,)32 b(ev)n(en)e(for)h Fu(X)j Fy(=)28 b Fr(R)2694 4086 y Fq(n)2745 4116 y Fy(,)j(there)g(is)g(no)f(extra)418 4215 y(e\013ort)h(to)f(go)g(from)g(the)h(Laplace)f(op)r(erator)f(to)h(a)h (general)e Fu(h)p Fy(\()p Fu(P)12 b Fy(\).)46 b(Ev)n(en)30 b(more)g(general)f(free)418 4315 y(Hamiltonians)20 b(are)g(a)n(v)-5 b(ailable)19 b(\(including)i(some)f(v)-5 b(ariable)20 b(co)r(e\016cien)n(t)h (di\013eren)n(tial)f(op)r(erators)418 4415 y(on)40 b Fr(R)600 4385 y Fq(n)651 4415 y Fy(\),)k(but)d(w)n(e)f(stic)n(k)g(to)g(the)h(case)e Fu(H)1784 4427 y Fs(0)1866 4415 y Fy(=)44 b Fu(h)p Fy(\()p Fu(P)12 b Fy(\))40 b(for)g(simplicit)n(y)-7 b(.)75 b(A)41 b(more)e(drastical) 418 4514 y(simpli\014cation)33 b(w)n(as)f(p)r(erformed)g(at)h(the)g(lev)n(el) g(of)f(p)r(oten)n(tials.)53 b(W)-7 b(e)33 b(decided)g(to)g(treat)f(only)418 4614 y(p)r(erturbations)23 b(whic)n(h)h(are)e(m)n(ultiplication)i(op)r (erators)e(b)n(y)h(b)r(ounded,)i(con)n(tin)n(uous)e(functions)418 4714 y Fu(V)44 b Fy(:)25 b Fu(X)32 b Ft(!)25 b Fr(R)p Fy(.)46 b(This)29 b(already)e(has)i(full)g(relev)-5 b(ance)28 b(for)g(the)i (formalism)e(and)g(the)h(phenomena)418 4813 y(w)n(e)c(w)n(an)n(t)f(to)h(put)g (in)n(to)g(evidence.)35 b(The)25 b(w)n(a)n(y)f(one)g(could)h(treat)g(more)f (general)f(\(higher-order,)418 4913 y(lo)r(cally)k(singular\))g(p)r (erturbations)f(is)i(v)n(ery)e(w)n(ell)i(illustrated)f(in)h([1],)f([3])h(and) f([6].)518 5013 y(In)33 b(the)h(\014rst)g(section)f(w)n(e)g(shall)g(presen)n (t)g(the)h(main)f(abstract)g(result,)i(a)e(general)f(HWZ-)418 5112 y(t)n(yp)r(e)c(form)n(ula)e(for)h(the)h(essen)n(tial)f(sp)r(ectrum.)37 b(In)27 b(Section)h(2)f(w)n(e)g(treat)g(a)g(class)g(of)g(p)r(oten)n(tials)p eop %%Page: 4 4 bop 418 314 a Fw(4)1111 b(MARIUS)28 b(M)1953 297 y(\025)1944 314 y(ANTOIU)418 513 y Fy(including)c(those)g(of)g(the)g(form)g Fu(V)42 b Fy(=)22 b Fu(V)1636 525 y Fs(1)1674 513 y Fu(V)1722 525 y Fs(2)1760 513 y Fy(,)j(where)e Fu(V)2092 525 y Fs(1)2154 513 y Fy(has)g(v)-5 b(anishing)24 b(oscillation)f(at)h(in\014nit)n(y)418 613 y(and)34 b Fu(V)634 625 y Fs(2)707 613 y Fy(is)g(almost)g(p)r(erio)r (dic.)57 b(This)34 b(case)g(is)g(a)g(signi\014can)n(t)g(extension)g(of)g ([8].)57 b(In)35 b(Section)418 712 y(3)h(w)n(e)g(tak)n(e)g(in)n(to)g(accoun)n (t)g(cartesian)f(anisotrop)n(y)-7 b(.)62 b(An)37 b(app)r(endix)g(is)f(dev)n (oted)g(to)g(graded)418 812 y(C*-algebras,)23 b(whic)n(h)j(pla)n(y)f(an)h (imp)r(ortan)n(t)f(role)g(in)h(the)g(N-b)r(o)r(dy)g(problem;)g(it)g(is)g(sho) n(wn)f(ho)n(w)418 911 y(they)j(\014t)g(in)g(our)f(general)f(framew)n(ork.)35 b(Eac)n(h)26 b(section)h(will)h(start)f(with)h(a)g(short)e(abstract.)518 1011 y(W)-7 b(e)19 b(men)n(tion)f(that)h(the)g(formalism)e(emplo)n(y)n(ed)h (in)h(this)f(article)g(ma)n(y)g(also)f(b)r(e)i(used)g(for)f(other)418 1111 y(purp)r(oses)27 b(than)g(the)h(study)g(of)f(the)h(essen)n(tial)e(sp)r (ectrum.)37 b(In)28 b([2])f(lo)r(calization)g(prop)r(erties)f(of)418 1210 y(anisotropic)39 b(Sc)n(hr\177)-42 b(odinger)38 b(op)r(erators)g(are)h (pro)n(v)n(en)g(b)n(y)h(lo)r(oking)f(at)h(ideals)f(asso)r(ciated)g(to)418 1310 y(in)n(v)-5 b(arian)n(t,)27 b(closed)g(subsets)g(of)g(a)h(suitable)f (dynamical)g(system.)1038 1510 y Fz(1.)45 b(Decompositions)31 b(in)f(the)h(Calkin)f(algebra)518 1660 y Fy(In)24 b(this)g(section)g(w)n(e)g (shall)f(pro)n(v)n(e)g(our)g(main)h(abstract)f(result.)35 b(T)-7 b(o)24 b("calculate")e(the)j(essen-)418 1759 y(tial)g(sp)r(ectrum)g(of)h(the) f(self-adjoin)n(t)g(op)r(erator)e Fu(H)32 b Fy(acting)25 b(in)g Ft(H)f Fy(=)f Fu(L)2594 1729 y Fs(2)2630 1759 y Fy(\()p Fu(X)7 b Fy(\),)26 b(w)n(e)f(need)g(to)g(sho)n(w)418 1859 y(that)g(its)f(resolv)n (en)n(ts)e Ft(f)p Fy(\()p Fu(H)c Ft(\000)11 b Fu(\020)6 b Fy(\))1396 1829 y Fn(\000)p Fs(1)1486 1859 y Ft(g)1528 1874 y Fq(\020)11 b(=)-41 b Fn(2)o Fl(R)1677 1859 y Fy(b)r(elong)23 b(to)h(some)g (C*-subalgebra)d Fk(C)j Fy(of)g Fu(B)t Fy(\()p Ft(H)q Fy(\))h(and)f(to)418 1959 y(express)g(its)i(image)e(in)i(the)g(Calkin)f(algebra)e(\(i.e.)37 b(the)26 b(quotien)n(t)f(of)g Fk(C)h Fy(b)n(y)f(the)h(ideal)f Fu(K)6 b Fy(\()p Ft(H)q Fy(\))25 b(of)418 2058 y(compact)30 b(op)r(erators\))g(in)h(an)f(e\016cien)n(t)h(w)n(a)n(y)-7 b(.)46 b(The)31 b(natural)f(C*-algebra)e(will)j(b)r(e)g(generated)418 2158 y(b)n(y)24 b(the)g(p)r(oten)n(tials)g(and)g(the)g(resolv)n(en)n(ts)f(of) h(the)g(kinetic)g(energies.)35 b(By)24 b(using)f(tec)n(hniques)h(in-)418 2258 y(v)n(olving)c(crossed)g(pro)r(ducts)h(w)n(e)g(will)h(b)r(e)g(reduced)f (to)g(understanding)g(the)g(dynamical)g(system)418 2361 y Ft(X)41 b Fy(:=)29 b(\()691 2340 y(^)667 2361 y Fu(X)e Ft(n)21 b Fu(X)r(;)14 b(\013;)g(X)7 b Fy(\).)47 b(Here)1425 2340 y(^)1402 2361 y Fu(X)37 b Fy(is)31 b(a)g(compacti\014cation)f(of)h Fu(X)38 b Fy(whic)n(h)31 b(tak)n(es)f(in)n(to)h(accoun)n(t)418 2460 y(the)d(b)r(eha)n(viour)f(at)h(in\014nit)n(y)g(of)g(the)h(p)r(oten)n(tials)e (and)h(on)g(whic)n(h)g(the)g(action)g(b)n(y)f(translations)418 2560 y Fu(\013)c Fy(of)f Fu(X)29 b Fy(on)22 b(itself)g(extends)g(con)n(tin)n (uously)-7 b(.)35 b(W)-7 b(e)22 b(in)n(tro)r(duce)g(the)h(cen)n(tral)e (concept)h(of)g(in)n(v)-5 b(arian)n(t)418 2660 y(decomp)r(osition)32 b(of)g(a)g(dynamical)f(system)h(and)g(apply)g(it)h(to)f Ft(X)12 b Fy(.)51 b(It)33 b(will)f(lead)g(straigh)n(tfor-)418 2759 y(w)n(ardly)g(to)h(an)g(em)n(b)r(edding)g(of)g Fk(C)p Fu(=K)6 b Fy(\()p Ft(H)q Fy(\))32 b(in)n(to)h(a)g(direct)g(pro)r(duct)g(of)g (C*-algebras.)50 b(These)418 2859 y(C*-algebras)21 b(are)i(canonically)g (de\014ned)i(b)n(y)f(the)g(mem)n(b)r(ers)g(of)g(the)h(in)n(v)-5 b(arian)n(t)23 b(decomp)r(osition)418 2958 y(and)28 b(they)h(con)n(tain)f (the)h(resolv)n(en)n(ts)e(of)h(the)h(asymptotic)f(Hamiltonians.)39 b(As)29 b(a)f(simple)h(con-)418 3058 y(sequence,)i(the)f(essen)n(tial)g(sp)r (ectrum)g(of)h Fu(H)37 b Fy(will)30 b(b)r(e)h(equal)f(to)g(the)h(closure)e (of)h(the)h(union)f(of)418 3158 y(the)e(sp)r(ectra)f(of)h(these)f(asymptotic) g(Hamiltonians.)418 3308 y FA([1.1])j(C*-algebras)i(and)g(the)g(essen)m(tial) h(sp)s(ectrum.)518 3407 y Fy(F)-7 b(or)27 b(an)n(y)g(Hilb)r(ert)i(space)e Ft(H)q Fy(,)i(w)n(e)e(set)h Fu(B)t Fy(\()p Ft(H)q Fy(\))h(for)f(the)g (C*-algebra)d(of)j(all)g(b)r(ounded,)h(linear)418 3507 y(op)r(erators)22 b(:)h Ft(H)h(!)f(H)q Fy(.)36 b(The)23 b(ideal)h(of)f(compact)h(op)r(erators)d (in)j Ft(H)h Fy(will)f(b)r(e)g(denoted)g(b)n(y)f Fu(K)6 b Fy(\()p Ft(H)q Fy(\).)418 3607 y(F)-7 b(rom)27 b(no)n(w)g(on,)h("ideal")e(will)i(alw) n(a)n(ys)e(mean)h("t)n(w)n(o-sided,)g(closed)g(ideal".)36 b(It)28 b(is)g(kno)n(wn)f(ho)n(w)418 3706 y(to)g(express)f(the)i(sp)r(ectrum)g(and)f (the)g(essen)n(tial)g(sp)r(ectrum)g(of)g(a)g(self-adjoin)n(t)g(op)r(erator)f Fu(H)34 b Fy(in)418 3806 y Ft(H)f Fy(in)f(terms)g(of)g(its)g(functional)g (calculus)f(with)i(functions)f(from)g Fu(C)2605 3818 y Fs(0)2642 3806 y Fy(\()p Fr(R)q Fy(\))k(:=)30 b Ft(f)p Fu(f)39 b Fy(:)30 b Fr(R)36 b Ft(!)31 b Fr(C)51 b Ft(j)418 3906 y Fu(f)36 b Fy(con)n(tin)n (uous)o Fu(;)42 b(f)9 b Fy(\()p Fu(\025)p Fy(\))23 b Ft(!)h Fy(0)j(when)h Ft(j)p Fu(\025)p Ft(j)23 b(!)g(1g)p Fy(:)772 4105 y Fu(\033)-51 b(\033)766 4101 y(\033)816 4105 y Fy(\()p Fu(H)7 b Fy(\))24 b(:=)e Ft(f)p Fu(\025)h Ft(2)h Fr(R)29 b Ft(j)50 b Fy(if)29 b Fu(f)i Ft(2)24 b Fu(C)1725 4117 y Fs(0)1762 4105 y Fy(\()p Fr(R)p Fy(\))34 b(and)28 b Fu(f)9 b Fy(\()p Fu(\025)p Fy(\))23 b Ft(6)p Fy(=)g(0)p Fu(;)41 b Fy(then)28 b Fu(f)9 b Fy(\()p Fu(H)e Fy(\))23 b Ft(6)p Fy(=)g(0)p Ft(g)p Fu(;)643 4464 y(\033)-51 b(\033)637 4460 y(\033)687 4476 y Fs(e)p Fq(ss)784 4464 y Fy(\()p Fu(H)7 b Fy(\))23 b(:=)g Ft(f)p Fu(\025)g Ft(2)g Fr(R)29 b Ft(j)51 b Fy(if)28 b Fu(f)k Ft(2)23 b Fu(C)1692 4476 y Fs(0)1730 4464 y Fy(\()p Fr(R)p Fy(\))34 b(and)27 b Fu(f)9 b Fy(\()p Fu(\025)p Fy(\))24 b Ft(6)p Fy(=)f(0)p Fu(;)41 b Fy(then)28 b Fu(f)9 b Fy(\()p Fu(H)e Fy(\))32 b Fu(=)-51 b Ft(2)23 b Fu(K)6 b Fy(\()p Ft(H)q Fy(\))p Ft(g)p Fu(:)518 4664 y Fy(W)-7 b(e)24 b(shall)f(extend)i(these)e(to)h(an)g(abstract)f (setting,)h(follo)n(wing)f([1].)35 b(W)-7 b(e)25 b(are)e(giv)n(en)g(an)g (ideal)418 4763 y Fk(K)28 b Fy(of)g(a)f(C*-algebra)e Fk(C)j Fy(not)f(necessarily)f(represen)n(ted)g(in)i(a)f(Hilb)r(ert)i(space.)418 4913 y FA(De\014nition)36 b(1.1.)47 b Fj(\(a\))31 b(A)g Fy(\(self-adjoin)n (t\))f(observ)-5 b(able)30 b(a\016liated)g(to)h Fk(C)f Fj(is)h(a)f (functional)h(cal-)418 5013 y(culus)h(taking)f(v)-5 b(alues)32 b(in)g Fk(C)p Fj(,)h(i.e.)50 b(a)32 b(*-morphism)e Fu(H)37 b Fy(:)30 b Fu(C)2264 5025 y Fs(0)2302 5013 y Fy(\()p Fr(R)p Fy(\))37 b Ft(!)30 b Fk(C)p Fj(.)50 b(Instead)32 b(of)f Fu(H)7 b Fy(\()p Fu(f)i Fy(\))32 b Fj(w)n(e)418 5112 y(shall)27 b(write)h (systematically)e Fu(f)9 b Fy(\()p Fu(H)e Fy(\))p Fj(.)p eop %%Page: 5 5 bop 1026 314 a Fw(C*{ALGEBRAS)29 b(AND)f(THE)g(ESSENTIAL)g(SPECTR)n(UM)601 b(5)518 513 y Fj(\(b\))28 b(The)g Fk(K)p Fj(-)p Fy(essen)n(tial)f(sp)r (ectrum)g Fj(of)h Fu(H)34 b Fj(is)746 723 y Fu(\033)-51 b(\033)740 719 y(\033)791 689 y Fi(C)791 744 y(K)837 723 y Fy(\()p Fu(H)7 b Fy(\))23 b(:=)g Ft(f)p Fu(\025)g Ft(2)g Fr(R)29 b Ft(j)51 b Fj(if)28 b Fu(f)k Ft(2)23 b Fu(C)1745 735 y Fs(0)1783 723 y Fy(\()p Fr(R)p Fy(\))34 b Fj(and)28 b Fu(f)9 b Fy(\()p Fu(\025)p Fy(\))23 b Ft(6)p Fy(=)g(0)p Fu(;)41 b Fj(then)28 b Fu(f)9 b Fy(\()p Fu(H)e Fy(\))32 b Fu(=)-51 b Ft(2)23 b Fk(K)p Ft(g)p Fu(:)518 932 y Fy(Ob)n(vious)30 b(imp)r(ortan)n(t)h(cases)f(are)g Fk(K)g Fy(=)f Ft(f)p Fy(0)p Ft(g)h Fy(\(w)n(e)h(write)g(simply)e Fu(\033)-51 b(\033)2565 928 y(\033)2615 902 y Fi(C)2615 959 y Fn(f)p Fs(0)p Fn(g)2721 932 y Fy(\()p Fu(H)7 b Fy(\))29 b Ft(\021)e Fu(\033)-51 b(\033)2976 928 y(\033)3026 902 y Fi(C)3069 932 y Fy(\()p Fu(H)7 b Fy(\)\))32 b(and)418 1052 y Fk(K)24 b Fy(=)e Fu(K)6 b Fy(\()p Ft(H)q Fy(\),)28 b(with)e Fu(\033)-51 b(\033)1028 1048 y(\033)1078 1022 y Fi(C)1078 1079 y Fq(K)t Fs(\()p Fn(H)p Fs(\))1251 1052 y Fy(\()p Fu(H)7 b Fy(\))24 b(=)c Fu(\033)-51 b(\033)1494 1048 y(\033)1544 1064 y Fs(ess)1631 1052 y Fy(\()p Fu(H)7 b Fy(\))28 b(for)f(an)n(y)g(C*-subalgebra)e Fk(C)i Fy(of)h Fu(B)t Fy(\()p Ft(H)q Fy(\))g(con)n(taining)418 1164 y Fu(K)6 b Fy(\()p Ft(H)q Fy(\).)518 1266 y(One)27 b(ma)n(y)g(transp)r (ort)f(observ)-5 b(ables)27 b(b)n(y)g(*-morphisms:)418 1423 y FA(De\014nition)k(1.2.)46 b Fj(Let)25 b Fu(\032)e Fy(:)h Fk(C)e Ft(!)i Fk(C)1548 1392 y Fn(0)1596 1423 y Fj(a)h(*-morphism)f(and)i Fu(H)32 b Fj(an)25 b(observ)-5 b(able)24 b(a\016liated)i(to)f Fk(C)p Fj(.)418 1522 y(Then)k Fu(\032)p Fy([)p Fu(H)7 b Fy(])24 b(:)h Fu(C)932 1534 y Fs(0)969 1522 y Fy(\()p Fr(R)p Fy(\))31 b Ft(!)25 b Fk(C)1277 1492 y Fn(0)1328 1522 y Fj(giv)n(en)j(b)n(y)g Fu(f)9 b Fy(\()p Fu(\032)p Fy([)p Fu(H)e Fy(]\))25 b(:=)f Fu(\032)p Fy([)p Fu(f)9 b Fy(\()p Fu(H)e Fy(\)])28 b Fj(is)h(an)f(observ)-5 b(able)27 b(a\016liated)h(to)418 1622 y Fk(C)469 1592 y Fn(0)492 1622 y Fj(,)g(called)f Fy(the)h(image)f(of)h Fu(H)34 b Fy(through)27 b Fu(\032)p Fj(.)518 1778 y Fy(W)-7 b(e)28 b(remark)e(that)i(*-morphisms)e (do)h(not)h(increase)e(the)i(sp)r(ectrum:)1574 1992 y Fu(\033)-51 b(\033)1568 1988 y(\033)1618 1958 y Fi(C)1660 1933 y Fh(0)1684 1992 y Fy(\()p Fu(\032)p Fy([)p Fu(H)7 b Fy(]\))23 b Ft(\032)e Fu(\033)-52 b(\033)2015 1988 y(\033)2066 1958 y Fi(C)2109 1992 y Fy(\()p Fu(H)7 b Fy(\))418 2198 y(and)28 b(one)f(ev)n(en)g(has)g(equalit)n (y)g(if)h Fu(\032)g Fy(is)f(a)g(monomorphism)g(\(=injectiv)n(e)g (*-morphism\).)518 2301 y(When)f(calculating)f(the)h(quotien)n(t)g(of)g(a)f (C*-algebra)e Fk(C)j Fy(b)n(y)f(an)h(ideal)f Fk(K)p Fy(,)i(it)f(ma)n(y)f(b)r (e)i(useful)418 2400 y(to)21 b("surround")f Fk(K)h Fy(with)h(a)f(family)h(of) f(larger)e(ideals)i(\()p Fk(K)2141 2412 y Fq(a)2182 2400 y Fy(\))2214 2412 y Fq(a)p Fn(2)p Fq(A)2349 2400 y Fy(.)35 b(In)22 b(practice)e(w)n(e)h(are)g(esp)r(ecially)418 2500 y(in)n(terested)j(in)h(the) g(case)f Fk(K)g Fy(=)e Fu(K)6 b Fy(\()p Ft(H)q Fy(\))25 b(and)f(it)i(will)e (b)r(e)h(desirable)f(that)h(the)g(C*-algebras)d Fk(C)p Fu(=)p Fk(K)3367 2512 y Fq(a)418 2599 y Fy(should)k(b)r(e)h("small)e(and)h(in)n (telligible".)36 b(It)27 b(is)f(also)f(useful)i(to)f(c)n(ho)r(ose)f(the)h (family)h(of)f(ideals)g(in)418 2699 y(an)h(economical)g(w)n(a)n(y)-7 b(.)418 2856 y FA(De\014nition)40 b(1.3.)47 b Fj(Let)34 b Fk(C)g Fj(b)r(e)h(a)e(C*-algebra)f(and)i Fk(K)g Fj(an)g(ideal)g(of)g Fk(C)p Fj(.)57 b(A)34 b(family)g(of)g(ideals)418 2955 y Fy(\()p Fk(K)505 2967 y Fq(a)546 2955 y Fy(\))578 2967 y Fq(a)p Fn(2)p Fq(A)741 2955 y Fj(will)28 b(b)r(e)g(called)f Fk(K)p Fj(-)p Fy(su\016cien)n(t)h Fj(if)518 3057 y(\(i\))g Ft(\\)688 3069 y Fq(a)p Fn(2)p Fq(A)823 3057 y Fk(K)878 3069 y Fq(a)942 3057 y Fy(=)23 b Fk(K)p Fj(,)518 3159 y(\(ii\))28 b Fk(K)711 3171 y Fq(a)779 3159 y Fj(is)g(not)g(con)n(tained)e(in)i Fk(K)1538 3171 y Fq(b)1600 3159 y Fj(if)g Fu(a)23 b Ft(6)p Fy(=)f Fu(b)p Fj(.)518 3261 y(When)28 b Fk(K)23 b Fy(=)g Ft(f)p Fy(0)p Ft(g)p Fj(,)j(instead)i(of)f Ft(f)p Fy(0)p Ft(g)p Fj(-su\016cien)n(t)f(w)n(e)h(sa)n (y)g(simply)h Fy(su\016cien)n(t)p Fj(.)518 3418 y Fy(W)-7 b(e)28 b(can)f(pass)g(v)n(ery)f(simply)i(from)f(a)g(general)f Fk(K)j Fy(to)e Fk(K)d Fy(=)e Ft(f)p Fy(0)p Ft(g)p Fy(.)418 3575 y FA(Lemma)36 b(1.4.)46 b Fj(Let)32 b Fy(\()p Fk(K)1197 3587 y Fq(a)1238 3575 y Fy(\))1270 3587 y Fq(a)p Fn(2)p Fq(A)1437 3575 y Fj(a)f Fk(K)p Fj(-su\016cien)n(t)h(family)f(of)h(ideals)f(in)h Fk(C)p Fj(.)49 b(Then)31 b Fk(J)3027 3587 y Fq(a)3097 3575 y Fy(:=)e Fk(K)3269 3587 y Fq(a)3310 3575 y Fu(=)p Fk(K)418 3674 y Fj(ma)n(y)g(b)r(e)h(iden)n(ti\014ed)h(with)f(an)g(ideal)f(of)h Fk(C)p Fu(=)p Fk(K)g Fj(and)f Fy(\()p Fk(J)2105 3686 y Fq(a)2146 3674 y Fy(\))2178 3686 y Fq(a)p Fn(2)p Fq(A)2343 3674 y Fj(is)g(a)h (su\016cien)n(t)g(family)g(of)f(ideals)418 3774 y(in)f Fk(C)p Fu(=)p Fk(K)p Fj(.)518 3930 y Fy(The)f(basic)g(tec)n(hnical)h(result)f(is)418 4087 y FA(Prop)s(osition)h(1.5.)47 b Fj(\(a\))24 b(Let)g Fy(\()p Fk(K)1489 4099 y Fq(a)1530 4087 y Fy(\))1562 4099 y Fq(a)p Fn(2)p Fq(A)1721 4087 y Fj(a)g Fk(K)p Fj(-su\016cien)n(t)g(family)g(of)h (ideals)e(of)h(the)h(C*-algebra)418 4187 y Fk(C)p Fj(.)37 b(Then)28 b(there)f(is)h(a)f(canonical)f(monomorphism)1592 4405 y Fk(C)p Fu(=)p Fk(K)d Fu(,)-14 b Ft(!)1890 4326 y Fg(Y)1878 4504 y Fq(a)p Fn(2)p Fq(A)2022 4405 y Fk(C)p Fu(=)p Fk(K)2170 4417 y Fq(a)2210 4405 y Fu(:)418 4694 y Fj(\(b\))31 b(Let)f Fu(H)38 b Fj(b)r(e)30 b(an)g(observ)-5 b(able)29 b(a\016liated)h(to)g Fk(C)p Fj(.)45 b(F)-7 b(or)29 b(an)n(y)h Fu(a)d Ft(2)h Fu(A)p Fj(,)j(let)g(us)f(set)g Fu(H)3038 4706 y Fq(a)3106 4694 y Ft(\021)d Fu(\031)3245 4706 y Fq(a)3285 4694 y Fy([)p Fu(H)7 b Fy(])418 4794 y Fj(the)28 b(image)f(of)g Fu(H)35 b Fj(b)n(y)27 b(the)h(canonical)f (surjection)g Fu(\031)2050 4806 y Fq(a)2113 4794 y Fy(:)c Fk(C)g Ft(!)g Fk(C)p Fu(=)p Fk(K)2487 4806 y Fq(a)2527 4794 y Fj(.)37 b(Then)1471 5034 y Fu(\033)-51 b(\033)1465 5030 y(\033)1516 4999 y Fi(C)1516 5054 y(K)1562 5034 y Fy(\()p Fu(H)7 b Fy(\))23 b(=)p 1813 4937 516 4 v 1832 4955 a Fg([)1813 5133 y Fq(a)p Fn(2)p Fq(A)1955 5034 y Fu(\033)-51 b(\033)1949 5030 y(\033)1999 5010 y Fi(C)p Fq(=)p Fi(K)2120 5018 y Fm(a)2155 5034 y Fy(\()p Fu(H)2256 5046 y Fq(a)2296 5034 y Fy(\))p Fu(:)p eop %%Page: 6 6 bop 418 314 a Fw(6)1111 b(MARIUS)28 b(M)1953 297 y(\025)1944 314 y(ANTOIU)418 513 y Fv(Pr)l(o)l(of.)44 b Fy(\(a\))26 b(The)g(k)n(ernel)f (of)g Fu(\031)1362 525 y Fq(a)1429 513 y Fy(is)g Fk(K)1565 525 y Fq(a)1606 513 y Fy(.)36 b(Th)n(us,)26 b(the)h(k)n(ernel)d(of)i(\()p Fu(\031)2453 525 y Fq(a)2494 513 y Fy(\))2526 525 y Fq(a)p Fn(2)p Fq(A)2684 513 y Fy(:)d Fk(C)g Ft(!)2910 451 y Fg(Q)2989 538 y Fq(a)p Fn(2)p Fq(A)3137 513 y Fk(C)p Fu(=)p Fk(K)3285 525 y Fq(a)3351 513 y Fy(is)418 613 y Fk(K)p Fy(.)518 718 y(\(b\))g(F)-7 b(ollo)n(ws)21 b(from)h(\(a\),)i(from)e(the)h(in)n(v)-5 b(ariance)21 b(of)i(the)f(sp)r(ectrum)h(under)f(monomorphisms)418 818 y(and)27 b(from)g(the)g(fact)h(that)f(the)h(sp)r(ectrum)f(of)g(an)g(observ)-5 b(able)25 b(a\016liated)i(to)g(a)g(direct)g(pro)r(duct)418 917 y(is)h(the)g(closure)e(of)i(the)g(union)f(of)h(the)g(sp)r(ectra)f(of)g (its)h(comp)r(onen)n(ts.)82 b Ff(\003)518 1084 y Fy(It)30 b(is)f(kno)n(wn)g (that)h(an)n(y)f(ideal)g Fk(K)h Fy(is)g(the)g(in)n(tersection)f(of)g(the)h (family)g Fk(M)p Fy(\()p Fk(K)p Fy(\))h(of)f(maximal)418 1184 y(ideals)f(con)n(taining)g(it)g(and)h(ob)n(viously)e Fk(M)p Fy(\()p Fk(K)p Fy(\))i(form)g(a)f(family)g(of)h Fk(K)p Fy(-su\016cien)n(t)f (ideals.)42 b(If)30 b(an)418 1284 y(ideal)f Fk(K)673 1296 y Fq(a)742 1284 y Fy(is)g(maximal,)f(then)h(the)g(quotien)n(t)g Fk(C)p Fu(=)p Fk(K)2001 1296 y Fq(a)2070 1284 y Fy(is)f(simple.)41 b(Unfortunatelly)-7 b(,)29 b(usually)f(w)n(e)418 1383 y(do)33 b(not)f(understand)h(the)g(set)g(of)f(all)h(\(maximal\))f(ideals)h(of)f(a)h (C*-algebra,)e(so)h(w)n(e)h(ha)n(v)n(e)e(to)418 1483 y(w)n(ork)c(only)h(with) h(those)f(whic)n(h)g(are)f(kno)n(wn)h(explicitly)-7 b(.)39 b(In)29 b(the)f(sequel)g(w)n(e)g(shall)g(sho)n(w)g(ho)n(w)418 1582 y(this)23 b(can)e(b)r(e)i(done)f(systematically)-7 b(,)22 b(in)h(a)e(con\014gurational)g(framew)n(ork,)g(for)h(a)f(rather)g(general)418 1682 y(class)27 b(of)g(C*-algebras.)418 1849 y FA([1.2])j(The)i(ab)s(elian)h (case;)f(top)s(ological)h(dynamical)h(systems.)518 1955 y Fy(In)e(the)h (examples)e(w)n(e)h(ha)n(v)n(e)f(in)h(mind,)i(the)f(relev)-5 b(an)n(t)31 b(C*-algebra)e Fk(C)k Fy(is)f(generated)f(b)n(y)g(a)418 2054 y(translational)d(in)n(v)-5 b(arian)n(t)28 b(ab)r(elian)g(C*-algebra)f Ft(C)33 b Fy(\(related)c(to)g(the)h(p)r(oten)n(tials\))f(and)f(a)h(v)n(ery) 418 2154 y(simple)23 b(C*-algebra)d(to)j(whic)n(h)g(the)g("free")f (Hamiltonians)h Fu(h)p Fy(\()p Fu(P)12 b Fy(\))23 b(are)f(a\016liated.)35 b(As)23 b(w)n(e)f(shall)418 2253 y(see)33 b(in)g FA([1.4])p Fy(,)f Fk(C)h Fy(ma)n(y)f(b)r(e)h(written)g(as)f(a)h(crossed)e(pro)r(duct)i Ft(C)27 b Fr(o)21 b Fu(X)39 b Fy(of)33 b(the)g(ab)r(elian)g(part)f Ft(C)418 2353 y Fy(b)n(y)25 b(the)g(group)f Fu(X)7 b Fy(.)36 b(Algebraic)24 b(op)r(erations)f(as)i(quotien)n(ts)f(and)h(\(restricted\))g (direct)g(pro)r(ducts)418 2453 y(are)33 b(distributiv)n(e)g(with)i(resp)r (ect)e(to)h(the)g(crossed)e(pro)r(duct,)j(so)e(it)h(is)g(useful)g(to)f(study) h(\014rst)418 2552 y(decomp)r(osition)h(prop)r(erties)f(at)i(the)g(ab)r (elian)f(lev)n(el)g(in)g(the)h(spirit)g(of)f(De\014nition)h(1.3.)60 b(W)-7 b(e)418 2652 y(will)36 b(place)f(ourselv)n(es)f(in)i(a)f(con)n(text)g (whic)n(h)h(is)f(relev)-5 b(an)n(t)35 b(for)g(our)g(purp)r(oses;)k(most)c(of) h(the)418 2752 y(assertions)26 b(ha)n(v)n(e)g(ob)n(vious)h(more)f(general)h (coun)n(terparts.)518 2857 y(Let)38 b(us)g(in)n(tro)r(duce)g(\014rst)g(some)g (notations.)68 b(If)39 b Fk(B)g Fy(is)f(a)g(C*-algebra)e(and)i Fu(Y)57 b Fy(a)38 b(lo)r(cally)418 2957 y(compact)30 b(space,)h(the)h(set)e (of)h(all)g(con)n(tin)n(uous)f(functions)h Fu(')d Fy(:)h Fu(Y)47 b Ft(!)29 b Fk(B)i Fy(whic)n(h)g(tend)g(to)g(0)f(at)418 3056 y(in\014nit)n(y)f(will)g(b)r(e)h(denoted)f(b)n(y)f Fu(C)1468 3068 y Fs(0)1506 3056 y Fy(\()p Fu(Y)19 b Fy(;)14 b Fk(B)p Fy(\).)41 b(It)29 b(is)g(a)f(C*-algebra)f(in)i(a)f(standard)g(w)n(a)n(y)-7 b(.)39 b(When)418 3156 y Fk(B)34 b Fy(=)g Fr(C)15 b Fy(,)42 b(w)n(e)33 b(write)h(simply)h Fu(C)1422 3168 y Fs(0)1459 3156 y Fy(\()p Fu(Y)19 b Fy(\).)57 b(If)34 b Fu(Y)53 b Fy(is)34 b(compact,)h(w)n(e)f(drop)g(the)g(index)g(0.)56 b Fu(B)t(C)6 b Fy(\()p Fu(Y)20 b Fy(\))418 3256 y(consists)32 b(of)h(con)n(tin)n(uous,)h (b)r(ounded)f(functions)h(on)f Fu(Y)18 b Fy(.)54 b(Those)32 b(whic)n(h)h(are)f(also)g(uniformly)418 3355 y(con)n(tin)n(uous)27 b(are)f(elemen)n(ts)i(of)f Fu(B)t(C)1532 3367 y Fq(u)1576 3355 y Fy(\()p Fu(Y)19 b Fy(\).)518 3461 y(W)-7 b(e)35 b(describ)r(e)f(no)n(w)g (our)f(framew)n(ork.)56 b(Let)35 b(us)f(denote)h(b)n(y)f(\()p Fu(X)r(;)14 b Fy(+\))34 b(an)h(ab)r(elian,)h(lo)r(cally)418 3560 y(compact)30 b(group.)43 b(W)-7 b(e)31 b(shall)e(consider)g(on)h Fu(X)37 b Fy(con)n(tin)n(uous)29 b(p)r(oten)n(tials)h(b)r(elonging)f(to)h(a)g (C*-)418 3660 y(algebra)c Ft(C)5 b Fy(.)37 b(It)28 b(will)g(b)r(e)g (necessary)e(to)h(mak)n(e)g(the)h(follo)n(wing)f(\(rather)g(natural\))g (assumptions)418 3759 y(on)f Ft(C)5 b Fy(:)36 b(\(a\))27 b Ft(C)k Fy(is)26 b(a)g(unital)h(C*-subalgebra)d(of)i Fu(B)t(C)2000 3771 y Fq(u)2044 3759 y Fy(\()p Fu(X)7 b Fy(\))27 b(con)n(taining)e Fu(C)2672 3771 y Fs(0)2710 3759 y Fy(\()p Fu(X)7 b Fy(\),)27 b(\(b\))g Ft(C)k Fy(is)26 b(stable)418 3859 y(under)k(translations,)f(i.e.)43 b(if)30 b Fu(')p Fy(\()p Ft(\001)p Fy(\))h(is)f(in)g Ft(C)5 b Fy(,)30 b(then)g Fu(')p Fy(\()p Ft(\001)21 b Fy(+)e Fu(y)s Fy(\))30 b(is)f(also)g(in)h Ft(C)35 b Fy(for)29 b(all)g Fu(y)h Ft(2)d Fu(X)7 b Fy(.)42 b(By)418 3967 y(standard)30 b(Gelfand)i(theory)-7 b(,)31 b(there)g(is)g(a)g(top)r(ological)e(dynamical)i(system)g Ft(X)41 b Fy(:=)29 b(\()3121 3946 y(^)3097 3967 y Fu(X)6 b(;)14 b(\013;)g(X)7 b Fy(\))418 4075 y(whic)n(h)28 b(is)g(canonically)e(asso)r (ciated)h(to)h Ft(C)5 b Fy(.)37 b(The)28 b(space)2195 4054 y(^)2171 4075 y Fu(X)35 b Fy(is)28 b(the)g(sp)r(ectrum)g(of)g Ft(C)k Fy(and)c(it)h(is)e(a)418 4174 y Fv(c)l(omp)l(acti\014c)l(ation)k Fy(of)f Fu(X)7 b Fy(,)29 b(i.e.)43 b(a)29 b(compact)g(space)g(in)h(whic)n(h)g Fu(X)36 b Fy(is)29 b(dense.)43 b(The)30 b(elemen)n(ts)f(of)418 4274 y Ft(C)36 b Fy(are)30 b(exactly)h(the)g(con)n(tin)n(uous)f(functions)i (on)f Fu(X)37 b Fy(whic)n(h)31 b(can)g(b)r(e)h(extended)f(con)n(tin)n(uously) 418 4377 y(on)559 4356 y(^)536 4377 y Fu(X)6 b Fy(,)30 b(so)g Ft(C)k Fy(and)c Fu(C)6 b Fy(\()1132 4356 y(^)1108 4377 y Fu(X)h Fy(\))30 b(are)f(naturally)g(isomorphic.)42 b(The)30 b(application)f Fu(\013)e Fy(:)g Fu(X)f Ft(\002)3144 4356 y Fy(^)3120 4377 y Fu(X)33 b Ft(!)3355 4356 y Fy(^)3332 4377 y Fu(X)418 4476 y Fy(is)h(con)n(tin)n(uous,)h(extends)g(the)g(group)e(la)n(w)g Fu(X)d Ft(\002)22 b Fu(X)41 b Ft(7!)34 b Fu(X)41 b Fy(and)34 b(satis\014es)g Fu(\013)2870 4488 y Fq(x)p Fs(+)p Fq(y)3033 4476 y Fy(=)g Fu(\013)3185 4488 y Fq(x)3249 4476 y Ft(\016)23 b Fu(\013)3367 4488 y Fq(y)418 4576 y Fy(for)33 b(all)g Fu(x;)14 b(y)36 b Ft(2)e Fu(X)7 b Fy(;)36 b(w)n(e)d(set)g Fu(\013)1373 4588 y Fq(x)1415 4576 y Fy(\()5 b(^)-47 b Fu(x)q Fy(\))33 b(:=)g Fu(\013)p Fy(\()p Fu(x;)20 b Fy(^)-48 b Fu(x)q Fy(\).)55 b(One)33 b(sa)n(ys)f(that)i Fu(\013)g Fy(is)g(an)f(action)g(of)g Fu(X)40 b Fy(b)n(y)418 4684 y(homeomorphisms)26 b(of)1193 4663 y(^)1169 4684 y Fu(X)34 b Fy(whic)n(h)27 b(extends)h(the)g(action)f(of)g Fu(X)34 b Fy(on)28 b(itself)g(b)n(y)f(translations.)518 4789 y(It)k(will)g(b)r(e)g(seen)f(in)h FA([1.4])d Fy(that)j(the)g(quotien)n(t)g Fk(C)p Fu(=K)6 b Fy(\()p Ft(H)q Fy(\))30 b(is)g(easy)g(to)g(express)g(if)h(w) n(e)f(ha)n(v)n(e)418 4897 y(informations)38 b(on)h(the)h(ab)r(elian)e (quotien)n(t)h Ft(C)5 b Fu(=C)1982 4909 y Fs(0)2019 4897 y Fy(\()p Fu(X)i Fy(\).)71 b(One)39 b(iden)n(ti\014es)g Ft(C)44 b Fy(with)39 b Fu(C)6 b Fy(\()3210 4876 y(^)3185 4897 y Fu(X)h Fy(\))40 b(as)418 5005 y(ab)r(o)n(v)n(e)33 b(and)h Fu(C)887 5017 y Fs(0)925 5005 y Fy(\()p Fu(X)7 b Fy(\))34 b(with)h(the)g(ideal)f(of)g (the)h(functions)g(in)g Fu(C)6 b Fy(\()2491 4984 y(^)2467 5005 y Fu(X)g Fy(\))35 b(whic)n(h)g(are)e(n)n(ull)h(on)h(the)418 5112 y(b)r(oundary)26 b Fu(Z)i Fy(:=)1011 5091 y(^)987 5112 y Fu(X)22 b Ft(n)15 b Fu(X)7 b Fy(.)36 b(It)26 b(follo)n(ws)f(that)i Ft(C)5 b Fu(=C)1958 5124 y Fs(0)1994 5112 y Fy(\()p Fu(X)i Fy(\))26 b(is)h(canonically)d(isomorphic)h(to)h Fu(C)6 b Fy(\()p Fu(Z)g Fy(\).)p eop %%Page: 7 7 bop 1026 314 a Fw(C*{ALGEBRAS)29 b(AND)f(THE)g(ESSENTIAL)g(SPECTR)n(UM)601 b(7)418 513 y Fy(In)34 b FA([1.3])d Fy(w)n(e)j(shall)f(apply)g(Prop)r (osition)f(1.5.)54 b(to)33 b(the)h(case)e Fk(C)h Fy(=)g Ft(C)38 b Fy(and)33 b Fk(K)h Fy(=)e Fu(C)3049 525 y Fs(0)3087 513 y Fy(\()p Fu(X)7 b Fy(\).)54 b(By)418 613 y(Lemma)30 b(1.4.)42 b(and)30 b(the)g(discussion)f(ab)r(o)n(v)n(e,)g(w)n(e)h(are)f(reduced)g(to)h Fk(C)c Fy(=)h Fu(C)6 b Fy(\()p Fu(Z)g Fy(\))30 b(and)g Fk(K)d Fy(=)f Ft(f)p Fy(0)p Ft(g)p Fy(.)418 712 y(T)-7 b(ec)n(hnical)31 b(reasons)e(connected)i(with)h(the)f(prop)r(erties)f(of)h(crossed)f(pro)r (ducts)h(obliges)f(us)h(to)418 812 y(lo)r(ok)25 b(for)f(su\016cien)n(t)i (families)f(comp)r(osed)g(of)g Fv(invariant)h Fy(ideals)f(in)h Fu(C)6 b Fy(\()p Fu(Z)g Fy(\).)37 b(Before)24 b(doing)h(this,)418 911 y(let)j(us)g(recall)e(some)h(general)f(facts)i(ab)r(out)f(top)r(ological) f(dynamical)h(systems.)518 1013 y(So,)i(let)h Ft(Y)j Fy(:=)26 b(\()p Fu(Y)5 b(;)14 b(\013;)g(X)7 b Fy(\))29 b(a)g(top)r(ological)f (dynamical)h(system.)42 b(W)-7 b(e)29 b(shall)g(alw)n(a)n(ys)f(assume)418 1113 y(that)e Fu(X)31 b Fy(is)25 b(an)g(ab)r(elian,)h(lo)r(cally)e(compact)h (group)f(and)h(that)h Fu(Y)44 b Fy(is)25 b(a)f(lo)r(cally)h(compact)g(space.) 418 1213 y(By)19 b(a)g(result)g(of)g(Ellis)g(\([5]\),)i(the)f(join)n(t)f(con) n(tin)n(uit)n(y)g(of)g Fu(\013)h Fy(follo)n(ws)e(from)h(its)h(separate)d(con) n(tin)n(uit)n(y)-7 b(.)418 1312 y(W)g(e)25 b(shall)e(in)n(terpret)h(the)h (map)f Fu(x)f Ft(7!)g Fu(\013)1639 1324 y Fq(x)1706 1312 y Fy(as)g(an)h(em)n(b)r(edding)g(of)g(the)h(group)e Fu(X)31 b Fy(in)n(to)24 b(the)g(group)418 1412 y(of)k(all)f(the)h(homeomorphisms)e(of)i Fu(Y)18 b Fy(.)518 1514 y(Let)26 b(us)g(denote)f(b)n(y)h Fk(Y)g Fy(the)g(top)r(ology)f(of)h Fu(Y)44 b Fy(and)26 b(b)n(y)g Fk(Y)2253 1526 y Fq(\013)2326 1514 y Fy(the)g(family)g(of)g(all)f(in)n(v)-5 b(arian)n(t)25 b(sets)418 1614 y(b)r(elonging)g(to)g Fk(Y)p Fy(.)36 b(It)26 b(is)f(easy)f(to)i(sho)n(w)e(that)i Fk(Y)1918 1626 y Fq(\013)1990 1614 y Fy(is)g(also)e(a)h(top)r(ology)f(on)h Fu(Y)18 b Fy(,)26 b(strictly)f(coarser)418 1713 y(than)j Fk(Y)g Fy(as)f(so)r(on)f(as)h Fu(\013)h Fy(is)g(not)g(trivial.)36 b(W)-7 b(e)28 b(shall)f(talk)g(ab)r(out)h Fu(\013)p Fy(-)p Fv(op)l(en)g Fy(and)g Fu(\013)p Fy(-)p Fv(close)l(d)h Fy(sets.)518 1815 y(Man)n(y)d(familiar)h(notions)g(of)g(top)r(ological)f(dynamics)h(ma)n (y)f(b)r(e)i(reform)n(ulated)e(in)i(terms)f(of)418 1915 y(this)h(new)f(top)r (ology)-7 b(.)36 b(A)28 b(singleton)e Ft(f)p Fu(y)s Ft(g)h Fy(is)g Fu(\013)p Fy(-closed)g(i\013)h Fu(y)i Fy(is)d(a)g(\014xed)g(p)r(oin)n (t.)37 b(The)28 b(closure)e(of)418 2014 y(an)d(orbit)g(is)g(the)h Fu(\013)p Fy(-closure)e(of)i(a)f(p)r(oin)n(t;)i(this)e(is)g(called)g(usually) g Fv(a)j(quasi-orbit)p Fy(.)37 b(A)24 b(dynamical)418 2114 y(\(sub\)system)35 b(for)g(whic)n(h)f(the)h(space)g(is)f(a)h(quasi-orbit)e (is)i(called)f Fv(top)l(olo)l(gic)l(al)t(ly)40 b(tr)l(ansitive)p Fy(.)418 2214 y(An)24 b Fu(\013)p Fy(-closed)e(set)h(is)g(saturated)f(with)h (quasi-orbits.)34 b(Remark)22 b(that)i(the)f(set)g(of)g(all)g(the)g(quasi-) 418 2313 y(orbits)d(is)f Fv(not)h Fy(a)g(partition)g(of)g Fu(Y)e Fy(;)23 b(t)n(w)n(o)c(quasi-orbits)g(ma)n(y)g(o)n(v)n(erlap)f(or)h(ma)n(y)h (ev)n(en)f(b)r(e)i(con)n(tained)418 2413 y(strictly)26 b(one)f(in)n(to)h(the) g(other.)36 b(W)-7 b(e)26 b(sa)n(y)f(that)h(a)g(quasi-orbit)e(is)i Fv(maximal)h Fy(if)f(it)h(is)e(not)h(strictly)418 2513 y(con)n(tained)j(in)n (to)g(some)g(other)g(quasi-orbit.)41 b(In)30 b(terms)f(of)g Fk(Y)2355 2525 y Fq(\013)2403 2513 y Fy(,)h(it)g(is)f(the)h(closure)e(of)i(a) f(p)r(oin)n(t)418 2612 y(whic)n(h)e(is)h(not)f(con)n(tained)g(strictly)g(in)g (the)h(closure)e(of)h(a)g(p)r(oin)n(t.)37 b(A)28 b(closed,)f(in)n(v)-5 b(arian)n(t)26 b(subset)418 2712 y Fu(Y)466 2724 y Fs(0)538 2712 y Fy(is)35 b Fv(minimal)g Fy(i\013)h(ev)n(ery)d(p)r(oin)n(t)i(in)g(it)g (is)f Fu(\013)p Fy(-dense.)58 b(Closed,)36 b(minimal)f(in)n(v)-5 b(arian)n(t)33 b(subsets)418 2811 y(are)h(quasi-orbits)e(whic)n(h)j(do)f(not) g(con)n(tain)g(strictly)g(some)g(other)g(quasi-orbit.)55 b(If)35 b Fu(Y)53 b Fy(itself)418 2911 y(is)33 b(minimal,)h(w)n(e)e(sa)n(y)f(that)i (the)g(dynamical)f(system)h Ft(Y)40 b Fy(is)32 b(minimal;)k(of)c(course,)h (it)g(will)g(b)r(e)418 3011 y(top)r(ologically)26 b(transitiv)n(e.)36 b Ft(Y)f Fy(is)27 b(minimal)h(i\013)g Fk(Y)1973 3023 y Fq(\013)2044 3011 y Fy(=)23 b Ft(f;)p Fu(;)14 b(Y)j Ft(g)p Fy(.)418 3167 y FA([1.3])30 b(In)m(v)-5 b(arian)m(t)35 b(decomp)s(ositions.)418 3324 y(De\014nition)f(1.6.)46 b Fj(An)30 b Fy(in)n(v)-5 b(arian)n(t)27 b(decomp)r(osition)h Fj(of)h(the)g(top)r(ological)e(dynamical)h(system)418 3423 y Ft(Y)41 b Fy(:=)34 b(\()p Fu(Y)5 b(;)14 b(\013;)g(X)7 b Fy(\))34 b Fj(is)g(an)g(appro)n(ximate)f(economical)g(co)n(v)n(ering)f(of) 68 b Fu(Y)53 b Fj(b)n(y)34 b(closed,)h(in)n(v)-5 b(arian)n(t)418 3523 y(sets,)28 b(i.e.)37 b(a)27 b(family)g Ft(D)f Fy(:=)c(\()p Fu(Y)1349 3535 y Fq(a)1390 3523 y Fy(\))1422 3535 y Fq(a)p Fn(2)p Fq(A)1585 3523 y Fj(with)28 b Fu(Y)37 b Ft(n)18 b Fu(Y)1967 3535 y Fq(a)2031 3523 y Ft(2)23 b Fk(Y)2178 3535 y Fq(\013)2225 3523 y Fj(,)28 b(suc)n(h)f(that)518 3625 y(\(i\))h Ft([)688 3637 y Fq(a)p Fn(2)p Fq(A)823 3625 y Fu(Y)871 3637 y Fq(a)939 3625 y Fj(is)g(dense)f(in)h Fu(Y)19 b Fj(,)518 3727 y(\(ii\))28 b(If)g Fu(a)23 b Ft(6)p Fy(=)g Fu(b)k Fj(then)h Fu(Y)1194 3739 y Fq(a)1262 3727 y Fj(is)g(not)f(a)g(subset)h(of)g Fu(Y)1960 3739 y Fq(b)1993 3727 y Fj(.)518 3829 y(The)f(set)h(of)g(all)f(in)n(v)-5 b(arian)n(t)26 b(decomp)r(ositions)h(of)h Ft(Y)35 b Fj(will)27 b(b)r(e)h(denoted)g(b)n(y)f Fr(D)14 b Fy(\()q Ft(Y)7 b Fy(\))p Fj(.)518 3991 y Fy(If)621 3970 y(~)602 3991 y Ft(D)31 b Fy(satis\014es)e (\(i\),)h(b)n(y)e(dropping)g(the)i(sets)e Fu(Y)1969 4003 y Fq(a)2039 3991 y Fy(whic)n(h)g(are)g(prop)r(erly)g(con)n(tained)g(in)i(some) 418 4097 y(other)38 b(sets)f(of)943 4076 y(~)923 4097 y Ft(D)s Fy(,)j(w)n(e)e(get)g(a)g(true)f(in)n(v)-5 b(arian)n(t)37 b(decomp)r(osition.) 68 b(F)-7 b(rom)38 b(no)n(w)f(on)h(w)n(e)f(shall)418 4197 y(assume)27 b(that)h Fu(Y)46 b Fy(is)28 b(compact.)418 4353 y FA(Prop)s(osition)33 b(1.7.)46 b Fj(\(a\))29 b(There)f(is)g(a)g(one-to-one)f(corresp)r(ondence)g (b)r(et)n(w)n(een)h(\(i\))h(in)n(v)-5 b(arian)n(t)418 4453 y(decomp)r(ositions)27 b(of)g Ft(Y)7 b Fj(,)28 b(\(ii\))h(su\016cien)n(t)f (families)f(of)h(in)n(v)-5 b(arian)n(t)26 b(ideals)h(of)h Fu(C)6 b Fy(\()p Fu(Y)19 b Fy(\))p Fj(.)518 4555 y(\(b\))30 b(Let)g Ft(D)g(\021)c Fy(\()p Fu(Y)1074 4567 y Fq(a)1115 4555 y Fy(\))1147 4567 y Fq(a)p Fn(2)p Fq(A)1309 4555 y Ft(2)h Fr(D)14 b Fy(\()p Ft(Y)8 b Fy(\))p Fj(.)50 b(Then)30 b(there)f(exist)h(a)g(monomorphism)e Fu(i)2966 4567 y Fn(D)3051 4555 y Fy(:)f Fu(C)6 b Fy(\()p Fu(Y)19 b Fy(\))27 b Ft(!)418 4592 y Fg(Q)497 4679 y Fq(a)p Fn(2)p Fq(A)645 4654 y Fu(C)6 b Fy(\()p Fu(Y)790 4666 y Fq(a)831 4654 y Fy(\))p Fj(.)518 4757 y(\(c\))21 b(Elemen)n(ts)f Fy(\()p Fu(')1078 4769 y Fq(a)1118 4757 y Fy(\))1150 4769 y Fq(a)p Fn(2)p Fq(A)1306 4757 y Fj(of)h(the)f(image)g(of)h Fu(i)1879 4769 y Fn(D)1956 4757 y Fj(satisfy)f Fy(lim)2324 4769 y Fq(x)p Fn(!)p Fs(0)2479 4757 y Fy(sup)2604 4777 y Fq(a)2667 4757 y Ft(k)i Fu(\013)2784 4769 y Fq(a;x)2882 4757 y Fy([)p Fu(')2959 4769 y Fq(a)3000 4757 y Fy(])t Ft(\000)t Fu(')3150 4769 y Fq(a)3213 4757 y Ft(k)p Fy(=)g(0)p Fj(.)418 4856 y(W)-7 b(e)28 b(deneto)r(d)g(b)n(y)f Fu(\013)1045 4868 y Fq(a;x)1171 4856 y Fj(the)h(restriction)e(of)i Fu(\013)1859 4868 y Fq(x)1928 4856 y Fj(to)g Fu(Y)2078 4868 y Fq(a)2118 4856 y Fj(.)418 5013 y Fv(Pr)l(o)l(of.)44 b Fy(\(a\))25 b(follo)n(ws)f(from)h(the)h(w)n(ell-kno)n(wn)d(corresp)r(ondence)g(b)r(et)n (w)n(een)i(ideals)g(in)g(C\(Y\))h(and)418 5112 y(closed)i(sets)f(in)i(Y.)f (Sp)r(eci\014cally)-7 b(,)29 b(if)f Fu(Y)1612 5124 y Fs(0)1678 5112 y Fy(is)g(a)g(closed)f(subset)h(in)h Fu(Y)19 b Fy(,)28 b Fu(C)2613 5082 y Fq(Y)2652 5090 y Fe(0)2689 5112 y Fy(\()p Fu(X)7 b Fy(\))24 b(:=)f Ft(f)p Fu(')h Ft(2)g Fu(C)6 b Fy(\()p Fu(Y)20 b Fy(\))k Ft(j)p eop %%Page: 8 8 bop 418 314 a Fw(8)1111 b(MARIUS)28 b(M)1953 297 y(\025)1944 314 y(ANTOIU)418 513 y Fu(')p Ft(j)495 525 y Fq(Y)534 533 y Fe(0)594 513 y Fy(=)23 b(0)p Ft(g)k Fy(is)h(an)f(ideal)h(in)g Fu(C)6 b Fy(\()p Fu(Y)19 b Fy(\))28 b(and)g(there)g(are)e(no)i(others.)37 b(If)28 b Fu(Y)2555 525 y Fs(0)2616 513 y Fr($)23 b Fu(Y)2770 483 y Fn(0)2752 534 y Fs(0)2794 513 y Fy(,)28 b(then)g Fu(C)3099 483 y Fq(Y)3152 458 y Fh(0)3138 499 y Fe(0)3179 513 y Fy(\()p Fu(X)7 b Fy(\))23 b Fr($)418 629 y Fu(C)483 599 y Fq(Y)522 607 y Fe(0)559 629 y Fy(\()p Fu(X)7 b Fy(\),)26 b(b)n(y)g(Urysohn's)e(lemma.) 37 b(W)-7 b(e)26 b(ha)n(v)n(e)e Ft(\\)1933 641 y Fq(a)1974 629 y Fu(C)2039 599 y Fq(Y)2078 607 y Fm(a)2119 629 y Fy(\()p Fu(Y)18 b Fy(\))24 b(=)e Fu(C)p 2425 551 161 3 v 2425 599 a Fn([)2470 607 y Fm(a)2507 599 y Fq(Y)2546 607 y Fm(a)2586 629 y Fy(\()p Fu(Y)d Fy(\);)27 b(th)n(us)f Ft(\\)3003 641 y Fq(a)3043 629 y Fu(C)3108 599 y Fq(Y)3147 607 y Fm(a)3188 629 y Fy(\()p Fu(Y)19 b Fy(\))k(=)418 728 y Ft(f)p Fy(0)p Ft(g)37 b Fy(i\013)j Ft([)747 740 y Fq(a)787 728 y Fu(Y)835 740 y Fq(a)914 728 y Fy(is)f(dense)f(in)h Fu(Y)19 b Fy(.)70 b(It)39 b(is)f(easy)g(to)h(see)f(that) h Fu(Y)2400 740 y Fs(0)2476 728 y Fy(is)g(in)n(v)-5 b(arian)n(t)37 b(i\013)i Fu(C)3105 698 y Fq(Y)3144 706 y Fe(0)3181 728 y Fy(\()p Fu(Y)19 b Fy(\))39 b(is)418 828 y(in)n(v)-5 b(arian)n(t.)518 929 y(\(b\))20 b(W)-7 b(e)19 b(just)h(set)f Fu(i)1088 941 y Fn(D)1146 929 y Fy(\()p Fu(')p Fy(\))24 b(:=)e(\()p Fu(')p Ft(j)1507 941 y Fq(Y)1546 949 y Fm(a)1587 929 y Fy(\))1619 941 y Fq(a)p Fn(2)p Fq(A)1755 929 y Fy(.)34 b(Ob)n(viously)-7 b(,)19 b(it)h(is)f(an)g(injectiv)n(e)g(*-morphism)f(from)418 1029 y Fu(C)6 b Fy(\()p Fu(Y)19 b Fy(\))26 b(to)739 967 y Fg(Q)817 1054 y Fq(a)p Fn(2)p Fq(A)966 1029 y Fu(C)6 b Fy(\()p Fu(Y)1111 1041 y Fq(a)1152 1029 y Fy(\).)36 b(Alternativ)n(ely)-7 b(,)25 b(one)g(ma)n(y)f(com)n(bine)h(\(a\))g(with)g(Prop)r(osition)f(1.5.\(a\))418 1136 y(b)n(y)32 b(setting)g Fk(K)870 1148 y Fq(a)940 1136 y Fy(=)e Fu(C)1100 1106 y Fq(Y)1139 1114 y Fm(a)1180 1136 y Fy(\()p Fu(X)7 b Fy(\))32 b(and)f Fk(K)g Fy(=)f Ft(f)p Fy(0)p Ft(g)p Fy(.)48 b(The)32 b(equiv)-5 b(alence)31 b(of)h(the)g(t)n(w)n(o)g(approac)n (hes)d(is)418 1236 y(clear,)34 b(since)g Fu(C)6 b Fy(\()p Fu(Y)19 b Fy(\))p Fu(=C)1160 1206 y Fq(Y)1199 1214 y Fm(a)1239 1236 y Fy(\()p Fu(Y)g Fy(\))34 b(is)g(isomorphic)e(with)i Fu(C)6 b Fy(\()p Fu(Y)2260 1248 y Fq(a)2301 1236 y Fy(\))34 b(and)f(the)h(canonical) e(surjection)418 1336 y Fu(C)6 b Fy(\()p Fu(Y)19 b Fy(\))24 b Ft(7!)f Fu(C)6 b Fy(\()p Fu(Y)19 b Fy(\))p Fu(=C)1047 1306 y Fq(Y)1086 1314 y Fm(a)1127 1336 y Fy(\()p Fu(Y)f Fy(\))28 b(b)r(ecomes)g(the)g(op)r(eration)e(of)i(restriction)e(to)i Fu(Y)2772 1348 y Fq(a)2812 1336 y Fy(.)518 1437 y(\(c\))g(W)-7 b(e)28 b(ha)n(v)n(e)622 1641 y(sup)666 1707 y Fq(a)770 1641 y Ft(k)23 b Fu(\013)888 1653 y Fq(a;x)985 1641 y Fy([)p Fu(')1062 1653 y Fq(a)1103 1641 y Fy(])18 b Ft(\000)g Fu(')1281 1653 y Fq(a)1345 1641 y Ft(k)p Fy(=)k(sup)1518 1707 y Fq(a)1622 1641 y Ft(k)h Fu(\013)1740 1653 y Fq(a;x)1851 1641 y Fy([)p Fu(')p Ft(j)1951 1653 y Fq(Y)1990 1661 y Fm(a)2031 1641 y Fy(])c Ft(\000)f Fu(')p Ft(j)2233 1653 y Fq(Y)2272 1661 y Fm(a)2335 1641 y Ft(k)p Fy(=)p Ft(k)k Fu(\013)2559 1653 y Fq(x)2601 1641 y Fy([)p Fu(')p Fy(])d Ft(\000)f Fu(')23 b Ft(k\000)-33 b(\000)-19 b(\000)-33 b(!)2950 1692 y Fq(x)p Fn(!)p Fs(0)3138 1641 y Fy(0)p Fu(:)418 1886 y Fy(and)28 b(this)f(\014nishes)h(the)g(pro)r(of.)82 b Ff(\003)518 2040 y Fy(The)27 b(monomorphism)g Fu(i)1304 2052 y Fn(D)1389 2040 y Fy(is)g(rarely)f(surjectiv)n(e)h(ev)n(en)g(if)h Fu(A)g Fy(is)g(\014nite.)518 2141 y(It)d(is)f(often)h(useful)g(to)g(apply)f (Prop)r(osition)f(1.7.)35 b(for)24 b Ft(D)2248 2153 y Fs(max)2398 2141 y Fy(=)f(\()p Fu(Y)2566 2153 y Fq(a)2606 2141 y Fy(\))2638 2153 y Fq(a)p Fn(2)p Fq(A)2773 2141 y Fy(,)j(the)f(most)f(re\014ned)418 2241 y(in)n(v)-5 b(arian)n(t)24 b(decomp)r(osition,)h(for)g(whic)n(h)g(ev)n (ery)f(comp)r(onen)n(t)h Fu(Y)2379 2253 y Fq(a)2444 2241 y Fy(is)g(a)g(\(maximal\))g(quasi-orbit,)418 2340 y(i.e.)35 b Ft(Y)614 2352 y Fq(a)677 2340 y Fy(:=)23 b(\()p Fu(Y)868 2352 y Fq(a)909 2340 y Fu(;)14 b(\013;)g(X)7 b Fy(\))20 b(is)h(top)r(ologically)e (transitiv)n(e.)34 b(Recall)20 b(that)h(w)n(e)f(are)g(mainly)g(in)n(terested) 418 2448 y(in)32 b(the)f(case)f(when)i Fu(Y)50 b Fy(is)31 b(the)g(b)r (oundary)g Fu(Z)j Fy(=)1983 2427 y(^)1959 2448 y Fu(X)27 b Ft(n)20 b Fu(X)38 b Fy(of)31 b Fu(X)37 b Fy(in)n(to)31 b(some)g (compacti\014cation)442 2535 y(^)418 2556 y Fu(X)j Fy(whic)n(h)28 b(is)f(the)h(sp)r(ectrum)g(of)g(the)g(C*-algebra)d(of)i(p)r(oten)n(tials)h Ft(C)5 b Fy(.)36 b(Then)28 b(w)n(e)f(ma)n(y)g(sa)n(y)g(that)418 2655 y Fv(we)36 b(expr)l(esse)l(d)f(the)h(C*-algebr)l(a)f Ft(C)5 b Fu(=C)1624 2667 y Fs(0)1661 2655 y Fy(\()p Fu(X)i Fy(\))34 b Fv(by)h(me)l(ans)g(of)h(an)g(invariant)g(de)l(c)l(omp)l(osition)h(of)418 2755 y(the)30 b(dynamic)l(al)i(system)e("at)f(in\014nity")h Ft(Z)g Fy(=)23 b(\()p Fu(Z)q(;)14 b(\013;)g(X)7 b Fy(\))30 b Fv(into)g(top)l(olo)l(gic)l(al)t(ly)j(tr)l(ansitive)d(c)l(om-)418 2855 y(p)l(onents)p Fy(.)56 b(Note)34 b(that)g(when)g Ft(Y)41 b Fy(itself)35 b(is)e(top)r(ologically)g(transitiv)n(e,)h(no)g(decomp)r (osition)f(is)418 2954 y(p)r(ossible.)j(This)24 b(sho)n(ws)g(clearly)g(wh)n (y)g(going)g(from)g Ft(C)30 b Fy(to)24 b(the)h(quotien)n(t)g Ft(C)5 b Fu(=C)2818 2966 y Fs(0)2855 2954 y Fy(\()p Fu(X)i Fy(\))25 b(leads)f(often)418 3062 y(to)j(new)f(opp)r(ortunities.)37 b(The)26 b(initial)h(dynamical)f(system)h Ft(X)35 b Fy(=)23 b(\()2550 3041 y(^)2526 3062 y Fu(X)7 b(;)14 b(\013;)g(X)7 b Fy(\))26 b(is)h(top)r(ologically)418 3162 y(transitiv)n(e)i(b)n(y)g (de\014nition,)i(so,)e(in)h(order)e(to)i(ha)n(v)n(e)e(some)h(c)n(hance)g(to)h (mak)n(e)f(some)g(decomp)r(o-)418 3261 y(sition,)e(one)f(has)h(\014rst)f(to)h (eliminate)g(the)g("large")e(\(dense\))i(orbit)f Fu(X)7 b Fy(.)36 b(After)28 b(p)r(erforming)e(the)418 3361 y(crossed)g(pro)r(duct,)i(similar)f (assertions)f(ma)n(y)g(b)r(e)i(stated)g(ab)r(out)f(the)h(quotien)n(t)g Fk(C)p Fu(=K)6 b Fy(\()p Ft(H)q Fy(\).)418 3515 y FA([1.4])30 b(Crossed)h(pro)s(ducts.)518 3616 y Fy(Since)e(it)g(pla)n(ys)e(only)i(the)g (role)f(of)g(an)h(in)n(termediary)e(to)r(ol,)i(w)n(e)f(decided)h(to)g(adopt)f (a)g(v)n(ery)418 3716 y(pragmatic)g(p)r(oin)n(t)i(of)f(view)h(up)r(on)f(the)h (crossed)e(pro)r(duct)i(of)f(a)g(C*-algebra)e(b)n(y)i(a)g(group.)42 b(In)418 3816 y(fact)25 b(w)n(e)f(do)g(not)g(de\014ne)h(it,)h(but)f(just)g (giv)n(e)e(a)h(con)n(v)n(enien)n(t)g(description)g(and)g(en)n(umerate)g(some) 418 3915 y(of)29 b(its)g(prop)r(erties.)40 b(W)-7 b(e)29 b(shall)g(tak)n(e)f (pro\014t)h(of)g(all)f(the)i(sp)r(eci\014c)f(facilities)g(of)g(our)f(framew)n (ork)418 4015 y(\(ab)r(elian)35 b(ob)5 b(jects,)37 b(F)-7 b(ourier)34 b(transform,)h(etc\).)60 b(F)-7 b(or)34 b(the)i(general)d(theory)i(w)n(e)f (send)h(to)g([9].)418 4114 y(A)30 b(brief)g(surv)n(ey)f(of)g(the)i(facts)e (that)h(are)f(relev)-5 b(an)n(t)29 b(for)h(sp)r(ectral)f(analysis)f(ma)n(y)h (b)r(e)i(found)f(in)418 4214 y([6],)36 b(Section)e(3;)j(this)d(includes)g (the)h(precise)e(de\014nition)h(of)h(the)f(crossed)f(pro)r(duct)h(and)g(the) 418 4314 y(pro)r(ofs)h(of)g(most)g(of)h(the)f(assertions)f(w)n(e)h(shall)g (mak)n(e)f(in)i(the)g(sequel.)60 b(W)-7 b(e)36 b(hop)r(e)f(that)h(the)418 4413 y(presen)n(t)27 b(approac)n(h)f(will)i(b)r(e)g(con)n(v)n(enien)n(t)f (for)g(the)h(un)n(trained)g(reader)e(and)h(not)h(to)r(o)g(lo)r(ose)f(for)418 4513 y(the)h(exp)r(ert.)518 4614 y(W)-7 b(e)37 b(shall)g(call)f Fv(ab)l(elian)k(C*-dynamic)l(al)h(system)c Fy(a)f(triplet)i(\()p Ft(A)p Fu(;)14 b(\013;)g(X)7 b Fy(\),)40 b(where)c Ft(A)h Fy(is)g(an)418 4714 y(ab)r(elian)30 b(C*-algebra,)e Fu(X)36 b Fy(is)30 b(an)f(ab)r(elian,)h (lo)r(cally)g(compact)f(group)g(and)g Fu(\013)i Fy(is)f(a)f(con)n(tin)n(uous) 418 4813 y(action)38 b(of)h Fu(X)45 b Fy(b)n(y)38 b(*-automorphisms)f(of)h Ft(A)p Fy(.)71 b(By)38 b(Gelfand)h(theory)-7 b(,)41 b(there)d(is)h(a)f (bijectiv)n(e)418 4913 y(corresp)r(ondence)27 b(b)r(et)n(w)n(een)i(ab)r (elian)g(C*-dynamical)e(systems)i(and)f(top)r(ological)g(dynamical)418 5013 y(systems)h(as)f(in)i FA([1.2])p Fy(.)40 b(As)29 b(a)g(rule,)h(w)n(e)e (shall)h(denote)g(b)n(y)g(the)h(same)f(letter)g(the)h(action)e(of)i Fu(X)418 5112 y Fy(on)d(the)g(C*-algebra)d Ft(A)k Fy(and)e(on)h(its)g(sp)r (ectrum.)37 b(W)-7 b(e)27 b(endo)n(w)f(the)i(dual)e Fu(X)2772 5082 y Fq(])2830 5112 y Fy(\(the)h(space)f(of)h(all)p eop %%Page: 9 9 bop 1026 314 a Fw(C*{ALGEBRAS)29 b(AND)f(THE)g(ESSENTIAL)g(SPECTR)n(UM)601 b(9)418 513 y Fy(con)n(tin)n(uous)26 b(morphisms)h(:)c Fu(X)30 b Ft(!)23 b Fr(T)f Fy(:=)h Ft(f)p Fu(z)j Ft(2)d Fr(C)44 b Ft(j)28 b(j)23 b Fu(z)j Ft(j)p Fy(=)d(1)p Ft(g)p Fy(\))j(with)i(the)g(standard)e (structure;)418 613 y(it)i(b)r(ecomes)g(an)g(ab)r(elian,)f(lo)r(cally)g (compact)h(group.)36 b Fu(X)2199 582 y Fq(])2258 613 y Fy(is)27 b(compact)h(i\013)g Fu(X)35 b Fy(is)27 b(discrete.)38 b(An)418 712 y(imp)r(ortan)n(t)27 b(role)g(will)h(b)r(e)g(pla)n(y)n(ed)e(b)n(y)i(the)g (C*-algebra)d Fu(C)2240 724 y Fs(0)2277 712 y Fy(\()p Fu(X)2385 682 y Fq(])2416 712 y Fy(\).)518 812 y FA(\(A\))32 b Fy(Let)h(us)f(\014x)g (an)g(ab)r(elian)g(C*-dynamical)e(system)i(\()p Ft(A)p Fu(;)14 b(\013;)g(X)7 b Fy(\).)51 b(W)-7 b(e)33 b(denote)f(b)n(y)g Ft(A)21 b(\014)418 911 y Fu(C)477 923 y Fs(0)515 911 y Fy(\()p Fu(X)623 881 y Fq(])653 911 y Fy(\))32 b(the)g(algebraic)d(tensor)i(pro)r (duct)g(of)g Ft(A)h Fy(and)f Fu(C)2207 923 y Fs(0)2244 911 y Fy(\()p Fu(X)2352 881 y Fq(])2383 911 y Fy(\).)48 b(It)32 b(consists)f(of)g(\014nite)h(linear)418 1011 y(com)n(binations)h(of)g(elemen) n(ts)g(of)h(the)g(form)f Fu(')23 b Ft(\012)f Fu( )s Fy(,)35 b(with)f Fu(')f Ft(2)g(A)h Fy(and)f Fu( )j Ft(2)e Fu(C)2981 1023 y Fs(0)3018 1011 y Fy(\()p Fu(X)3126 981 y Fq(])3157 1011 y Fy(\).)55 b Fv(The)418 1111 y(cr)l(osse)l(d)31 b(pr)l(o)l(duct)e(of)i Ft(A)f Fv(by)h Fu(X)j Fy(is)27 b(a)g(C*-algebra)e(denoted)j(b)n(y)g Ft(A)18 b Fr(o)2554 1123 y Fq(\013)2620 1111 y Fu(X)29 b Ft(\021)23 b(A)c Fr(o)f Fu(X)7 b Fy(,)27 b(in)h(whic)n(h)418 1210 y Ft(A)20 b(\014)g Fu(C)648 1222 y Fs(0)685 1210 y Fy(\()p Fu(X)793 1180 y Fq(])824 1210 y Fy(\))30 b(is)g(a)f(dense)h(subspace.)42 b(The)30 b(precise)f(algebraic)f(and)i(top)r(ological)e(structures)418 1310 y(on)c Ft(C)15 b Fr(o)c Fu(X)30 b Fy(will)24 b(not)g(b)r(e)g(giv)n(en)f (here;)h(but)h(note)e(that)i(they)f(dep)r(end)g(on)g(the)g(action)f Fu(\013)h Fy(of)g Fu(X)30 b Fy(on)418 1410 y Ft(A)p Fy(.)37 b(One)27 b(has)g Fr(C)39 b(o)18 b Fu(X)29 b Fy(=)23 b Fu(C)1273 1422 y Fs(0)1310 1410 y Fy(\()p Fu(X)1418 1379 y Fq(])1449 1410 y Fy(\).)37 b(In)28 b(fact,)f(our)g Ft(A)18 b Fr(o)g Fu(X)34 b Fy(is)27 b(a)g(F)-7 b(ourier)26 b(transformed)h(v)n(ersion)418 1509 y(of)h(the)g(one)f(whic)n(h)g(app)r(ears)g(usually)g(in)h(the)g (literature.)518 1609 y FA(\(B\))g Fy(Let)g(us)g(consider)f(a)g(second)h(ab)r (elian)f(C*-dynamical)g(system)g(\()p Ft(A)2796 1579 y Fn(0)2820 1609 y Fu(;)14 b(\013)2910 1579 y Fn(0)2934 1609 y Fu(;)g(X)7 b Fy(\))27 b(and)h(a)g(*-)418 1708 y(morphism)19 b Fu(\026)k Fy(:)g Ft(A)g(!)g(A)1178 1678 y Fn(0)1221 1708 y Fy(whic)n(h)c(is)g Fu(X)7 b Fy(-co)n(v)-5 b(arian)n(t)17 b(\(i.e.)34 b Fu(\013)2203 1678 y Fn(0)2203 1729 y Fq(x)2247 1708 y Ft(\016)r Fu(\026)22 b Fy(=)g Fu(\026)r Ft(\016)r Fu(\013)2599 1720 y Fq(x)2679 1708 y Fy(for)c(all)h Fu(x)24 b Ft(2)f Fu(X)7 b Fy(\).)34 b(Then)418 1808 y(there)23 b(is)f(a)g(unique)h(*-morphism)e Fu(\026)1535 1820 y Fl(o)1614 1808 y Fy(:)j Ft(A)8 b Fr(o)g Fu(X)30 b Ft(!)23 b(A)2079 1778 y Fn(0)2111 1808 y Fr(o)8 b Fu(X)29 b Fy(suc)n(h)22 b(that)h Fu(\026)2689 1820 y Fl(o)2745 1808 y Fy([)p Fu(')8 b Ft(\012)g Fu( )s Fy(])24 b(=)f Fu(\026)p Fy([)p Fu(')p Fy(])8 b Ft(\012)g Fu( )s Fy(.)418 1908 y(Let)25 b(us)g(denote)g(b)n(y)g Ft(J)40 b Fy(the)26 b(k)n(ernel)e(of)h Fu(\026)p Fy(;)h(it)f(is)g(in)n(v)-5 b(arian)n(t)24 b(under)h(the)g(action)f Fu(\013)p Fy(.)37 b(Then)25 b Ft(J)k Fr(o)13 b Fu(X)418 2007 y Fy(ma)n(y)26 b(b)r(e)h(iden)n(ti\014ed)g (to)f(an)g(ideal)h(of)f Ft(A)17 b Fr(o)e Fu(X)33 b Fy(whic)n(h)27 b(coincides)f(with)h(the)g(k)n(ernel)e(of)i Fu(\026)3144 2019 y Fl(o)3200 2007 y Fy(.)36 b(One)418 2107 y(has)31 b(a)g(canonical)f (isomorphism)h([)p Ft(A)21 b Fr(o)g Fu(X)7 b Fy(])p Fu(=)p Fy([)p Ft(J)35 b Fr(o)21 b Fu(X)7 b Fy(])2166 2085 y Ft(\030)2166 2111 y Fy(=)2260 2107 y([)p Ft(A)p Fu(=)p Ft(J)16 b Fy(])21 b Fr(o)f Fu(X)38 b Fy(\(on)32 b Ft(A)p Fu(=)p Ft(J)47 b Fy(there)31 b(is)g(a)418 2207 y(natural)22 b(action)h(of)g Fu(X)7 b Fy(\))23 b(whic)n(h)g(sends)g(the)h(class)e(mo)r(dulo)h Ft(J)i Fr(o)10 b Fu(X)29 b Fy(of)23 b Fu(')10 b Ft(\012)g Fu( )25 b Fy(in)n(to)e(\()p Fu(')10 b Fy(+)g Ft(J)15 b Fy(\))10 b Ft(\012)g Fu( )s Fy(.)518 2306 y FA(\(C\))22 b Fy(W)-7 b(e)22 b(are)e(giv)n(en)h(no)n(w)g(a)g(family)h (\()p Ft(A)1739 2318 y Fq(a)1779 2306 y Fu(;)14 b(\013)1869 2318 y Fq(a)1910 2306 y Fu(;)g(X)7 b Fy(\))2055 2318 y Fq(a)p Fn(2)p Fq(A)2211 2306 y Fy(of)21 b(ab)r(elian)g(C*-dynamical)f(systems.)418 2406 y(W)-7 b(e)34 b(can)e(consider)g(the)i(natural)e(pro)r(duct)h(action)g (of)g Fu(X)40 b Fy(in)2381 2344 y Fg(Q)2459 2431 y Fq(a)p Fn(2)p Fq(A)2608 2406 y Ft(A)2674 2418 y Fq(a)2714 2406 y Fy(.)54 b(Unfortunately)-7 b(,)35 b(it)418 2517 y(is)29 b(not)h(alw)n(a)n(ys)d(con)n (tin)n(uous.)41 b(But)30 b(w)n(e)f(can)g(form)g(the)g(largest)f (C*-subalgebra)2977 2455 y Fg(Q)3055 2476 y Fn(0)3055 2542 y Fq(a)p Fn(2)p Fq(A)3204 2517 y Ft(A)3270 2529 y Fq(a)3340 2517 y Fy(of)418 2555 y Fg(Q)497 2642 y Fq(a)p Fn(2)p Fq(A)645 2617 y Ft(A)711 2629 y Fq(a)779 2617 y Fy(on)g(whic)n(h)f(con)n(tin)n(uit)n (y)g(is)h(restored:)733 2806 y Fg(Y)720 2984 y Fq(a)p Fn(2)p Fq(A)851 2818 y Fn(0)874 2885 y Ft(A)940 2897 y Fq(a)1004 2885 y Fy(:=)1114 2743 y Fg(\()1181 2885 y Fy(\()p Fu(')1267 2897 y Fq(a)1308 2885 y Fy(\))1340 2897 y Fq(a)p Fn(2)p Fq(A)1498 2885 y Ft(2)1589 2806 y Fg(Y)1577 2984 y Fq(a)p Fn(2)p Fq(A)1721 2885 y Ft(A)1787 2897 y Fq(a)1878 2815 y Fg(\014)1878 2864 y(\014)1968 2885 y Fy(lim)1957 2936 y Fq(x)p Fn(!)p Fs(0)2108 2885 y Fy(sup)2152 2951 y Fq(a)2256 2885 y Ft(k)22 b Fu(\013)2373 2897 y Fq(a;x)2471 2885 y Fy([)p Fu(')2548 2897 y Fq(a)2588 2885 y Fy(])d Ft(\000)f Fu(')2767 2897 y Fq(a)2830 2885 y Ft(k)p Fy(=)23 b(0)3002 2743 y Fg(\))3082 2885 y Fu(:)418 3164 y Fy(Then)36 b(there)g(exists)f(a)g(monomorphism)1773 3097 y Fg(\000)1811 3102 y(Q)1889 3122 y Fn(0)1889 3189 y Fq(a)p Fn(2)p Fq(A)2038 3164 y Ft(A)2104 3176 y Fq(a)2145 3097 y Fg(\001)2206 3164 y Fr(o)24 b Fu(X)43 b Ft(7!)2527 3102 y Fg(Q)2605 3189 y Fq(a)p Fn(2)p Fq(A)2740 3164 y Fy(\()p Ft(A)2838 3176 y Fq(a)2902 3164 y Fr(o)24 b Fu(X)7 b Fy(\))35 b(sending)418 3264 y(\()p Fu(')504 3276 y Fq(a)545 3264 y Fy(\))577 3276 y Fq(a)p Fn(2)p Fq(A)730 3264 y Ft(\012)18 b Fu( )31 b Fy(in)n(to)d(\()p Fu(')1153 3276 y Fq(a)1212 3264 y Ft(\012)18 b Fu( )s Fy(\))1384 3276 y Fq(a)p Fn(2)p Fq(A)1519 3264 y Fy(.)518 3364 y(W)-7 b(e)39 b(remark)e(that)i(the)f(image)g(of)h Fu(i)1697 3376 y Fn(D)1793 3364 y Fy(in)f(Prop)r(osition)f(1.7.)69 b(is)39 b(in)f(fact)h(comprised)f(in) 418 3401 y Fg(Q)497 3422 y Fn(0)497 3488 y Fq(a)p Fn(2)p Fq(A)645 3463 y Fu(C)6 b Fy(\()p Fu(Y)790 3475 y Fq(a)831 3463 y Fy(\).)518 3563 y FA(\(D\))29 b Fy(Let)f Ft(A)h Fy(b)r(e)f(a)g(C*-subalgebra)e(of)i Fu(B)t(C)1888 3575 y Fq(u)1932 3563 y Fy(\()p Fu(X)7 b Fy(\),)28 b(stable)g(under)g(translations.)38 b(W)-7 b(e)28 b(w)n(ork)418 3663 y(in)33 b(the)h(Hilb)r(ert)f(space)g Ft(H)g Fy(:=)e Fu(L)1467 3633 y Fs(2)1504 3663 y Fy(\()p Fu(X)7 b Fy(\))33 b(\(with)h(a)e(Haar)g (measure\))h(and)f(iden)n(tify)i Ft(A)f Fy(with)h(its)418 3762 y(image)c(in)g Fu(B)t Fy(\()p Ft(H)q Fy(\))h(formed)f(b)n(y)g(m)n (ultiplication)h(op)r(erators.)43 b(In)31 b FA([1.5])d Fy(op)r(erators)g Fu( )s Fy(\()p Fu(P)12 b Fy(\))31 b(in)g Ft(H)418 3862 y Fy(are)i(assigned)f (bijectiv)n(ely)i(to)f(a)g(large)f(class)h(of)g(functions)h Fu( )j Fy(de\014ned)d(on)f(the)h(dual)f(group)418 3962 y Fu(X)494 3932 y Fq(])524 3962 y Fy(.)j(In)23 b(particular,)g(w)n(e)g(get)g(a)g (canonical)g(iden)n(ti\014cation)g(of)g Fu(C)2408 3974 y Fs(0)2446 3962 y Fy(\()p Fu(X)2554 3932 y Fq(])2584 3962 y Fy(\))h(with)g(a)f (C*-subalgebra)418 4061 y(of)36 b Fu(B)t Fy(\()p Ft(H)q Fy(\),)i(denoted)e (in)g(the)g(same)f(w)n(a)n(y)-7 b(.)59 b(One)36 b(ma)n(y)e(form)2371 3994 y Fg(\012)2410 4061 y Ft(A)19 b(\001)f Fu(C)2595 4073 y Fs(0)2633 4061 y Fy(\()p Fu(X)2741 4031 y Fq(])2771 4061 y Fy(\))2803 3994 y Fg(\013)2843 4061 y Fy(,)38 b(the)e(closure)e(of)418 4169 y(the)39 b(v)n(ector)d(subspace)i(generated)f(b)n(y)g Ft(A)i Fy(and)f Fu(C)2039 4181 y Fs(0)2076 4169 y Fy(\()p Fu(X)2184 4139 y Fq(])2215 4169 y Fy(\).)69 b(It)38 b(can)g(b)r(e)g(sho)n(wn)g(that)g (it)g(is)g(a)418 4269 y(C*-subalgebra)d(and)i(that)h(it)f(coincides)g(with) 1986 4202 y Fg(\012)2025 4269 y Fu(C)2084 4281 y Fs(0)2122 4269 y Fy(\()p Fu(X)2230 4239 y Fq(])2260 4269 y Fy(\))19 b Ft(\001)g(A)2419 4202 y Fg(\013)2496 4269 y Fy(\(ob)n(vious)36 b(notation\).)66 b(The)418 4368 y(really)25 b(useful)h(fact)g(is)f(that)i Ft(A)15 b Fr(o)f Fu(X)32 b Fy(and)26 b Ft(hA)19 b(\001)f Fu(C)1941 4380 y Fs(0)1979 4368 y Fy(\()p Fu(X)7 b Fy(\))p Ft(i)26 b Fy(are)e(isomorphic,)h(the)i(elemen)n(t)e Fu(')16 b Ft(\012)e Fu( )418 4468 y Fy(corresp)r(onding)26 b(to)h Fu(')19 b Ft(\001)g Fu( )s Fy(\()p Fu(P)12 b Fy(\).)518 4568 y FA(\(E\))31 b Fy(With)i(the)e (notations)g(of)g FA(\(D\))p Fy(,)h(one)f(has)2028 4500 y Fg(\012)2068 4568 y Fu(C)2127 4580 y Fs(0)2164 4568 y Fy(\()p Fu(X)7 b Fy(\))18 b Ft(\001)h Fu(C)2423 4580 y Fs(0)2461 4568 y Fy(\()p Fu(X)2569 4538 y Fq(])2599 4568 y Fy(\))2631 4500 y Fg(\013)2700 4568 y Fy(=)29 b Fu(K)6 b Fy(\()p Ft(H)q Fy(\).)48 b(Th)n(us)32 b(the)418 4667 y(crossed)21 b(pro)r(duct)g Fu(C)1064 4679 y Fs(0)1102 4667 y Fy(\()p Fu(X)7 b Fy(\))g Fr(o)g Fu(X)28 b Fy(can)22 b(b)r(e)g(iden)n(ti\014ed)g(with)h(the)f(ideal)g(of)g(all)g (compact)f(op)r(erators)418 4767 y(in)28 b Fu(L)572 4737 y Fs(2)609 4767 y Fy(\()p Fu(X)7 b Fy(\).)418 4913 y FA([1.5])30 b(Generalized)j(Sc)m(hr\177)-48 b(odinger)34 b(op)s(erators)d(and)i(their)f (essen)m(tial)h(sp)s(ectrum.)518 5013 y Fy(In)d(this)h(paragraph)d(w)n(e)i (are)g(\014nally)g(in)g(a)h(p)r(osition)f(to)g(de\014ne)h(generalized)e(Sc)n (hr\177)-42 b(odinger)418 5112 y(op)r(erators)26 b(and)h(giv)n(e)g(a)g (result)g(on)h(their)f(essen)n(tial)g(sp)r(ectrum.)37 b(W)-7 b(e)28 b(shall)f(\014x)h(the)g(follo)n(wing)p eop %%Page: 10 10 bop 418 314 a Fw(10)1076 b(MARIUS)28 b(M)1953 297 y(\025)1944 314 y(ANTOIU)418 513 y FA(F)-8 b(ramew)m(ork.)48 b Fj(\(i\))29 b Fu(X)34 b Fj(will)27 b(b)r(e)h(an)g(ab)r(elian,)f(lo)r(cally)g(compact)g (group,)g(with)h(dual)f Fu(X)3175 483 y Fq(])3205 513 y Fj(.)518 615 y(\(ii\))32 b Ft(C)k Fj(will)c(b)r(e)g(an)g(unital)g(C*-subalgebra)c(of)k Fu(B)t(C)2155 627 y Fq(u)2199 615 y Fy(\()p Fu(X)7 b Fy(\))32 b Fj(whic)n(h)f(is)h(in)n(v)-5 b(arian)n(t)30 b(under)i(the)418 715 y(action)27 b Fu(\013)h Fj(of)g Fu(X)34 b Fj(b)n(y)27 b(left)h (translations.)518 821 y(\(iii\))e(W)-7 b(e)25 b(denote)g(b)n(y)1219 800 y Fy(^)1195 821 y Fu(X)32 b Fj(the)26 b(compact)e(sp)r(ectrum)i(of)f Ft(C)30 b Fj(and)25 b(b)n(y)g Fu(Z)30 b Fj(the)c(b)r(oundary)3188 800 y Fy(^)3164 821 y Fu(X)20 b Ft(n)14 b Fu(X)7 b Fj(.)418 928 y(The)32 b(extension)f(to)1093 907 y Fy(^)1069 928 y Fu(X)38 b Fj(of)32 b(the)g(action)f(of)g Fu(X)7 b Fj(,)32 b(as)f(w)n(ell)g(as)g(all)h (its)f(p)r(ossible)g(restrictions)g(will)418 1028 y(also)c(b)r(e)h(denoted)f (b)n(y)h Fu(\013)p Fj(.)518 1130 y(\(iv\))g Fu(h)23 b Fy(:)g Fu(X)870 1100 y Fq(])923 1130 y Ft(!)g Fr(R)34 b Fj(will)28 b(b)r(e)g(a)f(con)n(tin)n(uous)g(function,)h(satisfying)f(lim)2705 1142 y Fq(p)p Fn(!1)2876 1130 y Ft(j)p Fu(h)p Fy(\()p Fu(p)p Fy(\))p Ft(j)c Fy(=)g Ft(1)p Fj(.)518 1287 y Fy(W)-7 b(e)33 b(giv)n(e)e(a)i(sense)f(to)g(the)h(free)g(op)r(erator)e Fu(H)1962 1299 y Fs(0)2030 1287 y Fy(:=)g Fu(h)p Fy(\()p Fu(P)12 b Fy(\))33 b(in)g Ft(H)g Fy(:=)e Fu(L)2740 1257 y Fs(2)2776 1287 y Fy(\()p Fu(X)7 b Fy(\))33 b(b)n(y)f(using)h(the)418 1387 y(unitary)c(F)-7 b(ourier)29 b(transform)f Ft(F)35 b Fy(:)26 b Fu(L)1587 1357 y Fs(2)1624 1387 y Fy(\()p Fu(X)7 b Fy(\))26 b Ft(!)g Fu(L)1956 1357 y Fs(2)1993 1387 y Fy(\()p Fu(X)2101 1357 y Fq(])2132 1387 y Fy(\).)43 b(In)30 b Fu(L)2393 1357 y Fs(2)2429 1387 y Fy(\()p Fu(X)2537 1357 y Fq(])2568 1387 y Fy(\))g(w)n(e)f(consider)g(the)h (max-)418 1487 y(imal)h(op)r(erator)f Fu(M)1026 1499 y Fq(h)1099 1487 y Fy(of)h(m)n(ultiplication)h(b)n(y)f Fu(h)g Fy(and)g(w)n(e)f(set)i Fu(h)p Fy(\()p Fu(P)12 b Fy(\))29 b(:=)f Ft(F)2746 1457 y Fn(\000)p Fs(1)2836 1487 y Fu(M)2917 1499 y Fq(h)2959 1487 y Ft(F)8 b Fy(.)48 b(W)-7 b(ritting)418 1586 y Fu(H)487 1598 y Fs(0)558 1586 y Fy(as)32 b(a)h(function)h(of)g Fu(P)45 b Fy(\(men)n(t)34 b(to)f(b)r(e)h(the)g(momen)n(tum\))f(is)h(just)g(a)f(reminiscence)f(of)i(the) 418 1686 y(situations)27 b(in)h(whic)n(h)g(this)g(can)f(b)r(e)h(soundly)f (justi\014ed,)h Fu(X)i Fy(=)22 b Fr(R)2445 1656 y Fq(n)2524 1686 y Fy(for)27 b(instance.)518 1788 y(W)-7 b(e)37 b(shall)g(call)g Fv(gener)l(alize)l(d)j(Schr\177)-42 b(odinger)41 b(op)l(er)l(ator)d Fy(the)g(self-adjoin)n(t)f(op)r(erator)e Fu(H)46 b Fy(:=)418 1888 y Fu(H)487 1900 y Fs(0)547 1888 y Fy(+)23 b Fu(V)c Fy(,)37 b(where)d Fu(V)53 b Fy(is)35 b(the)g(op)r(erator)e(of)h(m)n(ultiplication)h (with)g(the)g(real-v)-5 b(alued)33 b(function)418 1987 y(\(also)d(denoted)h (b)n(y\))g Fu(V)19 b Fy(,)32 b(whic)n(h)e(w)n(e)h(c)n(ho)r(ose)e(in)i Ft(C)5 b Fy(.)47 b(This)31 b(function)g(will)g(b)r(e)g(\014xed)g(from)g(no)n (w)418 2095 y(on)c(and)h(its)g(con)n(tin)n(uous)e(extension)h(to)1719 2074 y(^)1695 2095 y Fu(X)34 b Fy(will)28 b(b)r(e)g(denoted)f(b)n(y)2509 2074 y(^)2497 2095 y Fu(V)18 b Fy(.)518 2197 y(The)27 b(p)r(oin)n(t)h(\(b\))h (of)e(the)h(next)g(result)f(is)h(our)e(main)i(abstract)f(con)n(tribution.)418 2355 y FA(Theorem)35 b(1.8.)46 b Fj(\(a\))31 b(The)g(self-adjoin)n(t)f(op)r (erator)f Fu(H)37 b Fj(is)31 b(a\016liated)f(to)h(the)g(C*-subalgebra)418 2454 y Fk(C)23 b Fy(:=)603 2387 y Fg(\012)642 2454 y Ft(C)g(\001)18 b Fu(C)809 2466 y Fs(0)847 2454 y Fy(\()p Fu(X)955 2424 y Fq(])986 2454 y Fy(\))1018 2387 y Fg(\013)1085 2454 y Fj(of)27 b Fu(B)t Fy(\()p Ft(H)q Fy(\))p Fj(.)518 2557 y(\(b\))35 b(Let)g Ft(D)i Fy(=)d(\()p Fu(Z)1108 2569 y Fq(a)1148 2557 y Fy(\))1180 2569 y Fq(a)p Fn(2)p Fq(A)1350 2557 y Fj(b)r(e)h(an)f(in)n(v)-5 b(arian)n(t)34 b(decomp)r(osition)g(of)g(the)h(dynamical)f(system)418 2656 y Ft(Z)c Fy(=)23 b(\()p Fu(Z)q(;)14 b(\013;)g(X)7 b Fy(\))p Fj(.)36 b(F)-7 b(or)24 b(an)n(y)g Fu(a)f Ft(2)h Fu(A)p Fj(,)i(let)f Fu(\032)1697 2668 y Fq(a)1760 2656 y Fy(:)e Fu(C)6 b Fy(\()p Fu(Z)1960 2668 y Fq(a)2001 2656 y Fy(\))23 b Ft(!)g Fu(B)t(C)2288 2668 y Fq(u)2332 2656 y Fy(\()p Fu(X)7 b Fy(\))25 b Fj(b)r(e)g(a)g (monomorphism.)35 b(W)-7 b(e)418 2781 y(set)28 b Fu(V)596 2793 y Fq(a)659 2781 y Fy(:=)23 b Fu(\032)813 2793 y Fq(a)867 2689 y Fg(h)919 2760 y Fy(^)906 2781 y Fu(V)c Ft(j)996 2793 y Fq(Z)1041 2801 y Fm(a)1082 2689 y Fg(i)1149 2781 y Fj(and)27 b Fu(H)1379 2793 y Fq(a)1442 2781 y Fy(:=)c Fu(H)1622 2793 y Fs(0)1678 2781 y Fy(+)18 b Fu(V)1809 2793 y Fq(a)1849 2781 y Fj(.)37 b(Then)418 3054 y Fy(\(1-1\))944 b Fu(\033)-51 b(\033)1532 3050 y(\033)1582 3066 y Fs(ess)1669 3054 y Fy(\()p Fu(H)7 b Fy(\))23 b(=)p 1920 2957 342 4 v 1939 2976 a Fg([)1920 3154 y Fq(a)p Fn(2)p Fq(A)2062 3054 y Fu(\033)-51 b(\033)2056 3050 y(\033)2106 3054 y Fy([)p Fu(H)2198 3066 y Fq(a)2239 3054 y Fy(])p Fu(:)518 3347 y FA(Remark.)36 b Fy(W)-7 b(e)27 b(assumed)f(that)h(the) g(ab)r(elian)g(C*-algebras)c Fu(C)6 b Fy(\()p Fu(Z)2600 3359 y Fq(a)2641 3347 y Fy(\))27 b(are)f(faithfully)h(repre-)418 3447 y(sen)n(ted)22 b(in)f Ft(H)i Fy(b)n(y)e(m)n(ultiplication)h(op)r (erators)e(b)r(ecause)h(this)h(really)e(happ)r(ens)i(in)g(the)g(examples)418 3546 y(w)n(e)k(ha)n(v)n(e)e(in)i(mind.)37 b(If)26 b(this)g(is)g(not)g(the)g (case,)f(there)h(is)f(a)h(less)f(concrete)g(v)n(ersion)f(of)i(Theorem)418 3646 y(1.8.,)f(in)g(whic)n(h)f(the)i(asymptotic)e(Hamiltonian)g Fu(H)2032 3658 y Fq(a)2098 3646 y Fy(is)g(de\014ned)h(as)f(observ)-5 b(ables)24 b(a\016liated)g(to)418 3746 y(the)k(abstract)f(C*-algebra)e Fu(C)6 b Fy(\()p Fu(Z)1459 3758 y Fq(a)1499 3746 y Fy(\))19 b Fr(o)f Fu(X)7 b Fy(.)518 3848 y FA(Remark.)52 b Fy(Muc)n(h)33 b(larger)e(classes)g(of)i(Hamiltonians)f(ma)n(y)g(b)r(e)h(treated)g(b)n(y)f (using)h(more)418 3947 y(so\014sticated)27 b(a\016liation)g(criteria,)f(as)h (those)h(in)f([1])h(or)f([3].)418 4105 y Fv(Pr)l(o)l(of)32 b(of)g(The)l(or)l(em)f(1.8.)45 b Fy(\(a\))28 b(It)h(is)g(pro)n(v)n(en)e(in)i ([1])f(that)h Fu(H)36 b Fy(is)28 b(a\016liated)h(to)f Fk(C)h Fy(if)g(\()p Fu(H)d Ft(\000)19 b Fu(\020)6 b Fy(\))3317 4075 y Fn(\000)p Fs(1)418 4204 y Fy(is)28 b(in)f Fk(C)h Fy(for)f(some)g(n)n(um)n (b)r(er)g Fu(\020)39 b(=)-51 b Ft(2)23 b Fr(R)p Fy(.)43 b(W)-7 b(e)28 b(c)n(ho)r(ose)f Fu(\020)34 b Fy(suc)n(h)27 b(that)h(the)g(Neumann)g (series)418 4424 y(\(1-2\))376 b(\()p Fu(H)26 b Ft(\000)18 b Fu(\020)6 b Fy(\))1254 4389 y Fn(\000)p Fs(1)1367 4424 y Fy(=)1467 4345 y Fg(X)1455 4523 y Fq(m)p Fn(2)p Fl(N)1600 4424 y Fy(\()p Ft(\000)p Fy(1\))1771 4389 y Fq(m)1834 4424 y Fy(\()p Fu(H)1935 4436 y Fs(0)1991 4424 y Ft(\000)18 b Fu(\020)6 b Fy(\))2148 4389 y Fn(\000)p Fs(1)2252 4356 y Fg(\002)2286 4424 y Fu(V)19 b Fy(\()p Fu(H)2454 4436 y Fs(0)2510 4424 y Ft(\000)f Fu(\020)6 b Fy(\))2667 4389 y Fn(\000)p Fs(1)2757 4356 y Fg(\003)2792 4372 y Fq(m)418 4714 y Fy(con)n(v)n(erges.)33 b(W)-7 b(e)23 b(ha)n(v)n(e)f(to)h(pro)n(v)n(e)f(that)h(eac)n(h)g(term)g(in)g(\(1-2\))g(is)g (in)g Fk(C)p Fy(.)36 b(By)22 b(taking)h(in)n(to)g(accoun)n(t)418 4813 y(the)34 b(de\014nition)g(of)f Fk(C)h Fy(and)f(the)h(fact)f(that)h Fu(V)53 b Fy(is)33 b(in)h Ft(C)5 b Fy(,)34 b(w)n(e)g(see)f(that)g(it)h(is)g (enough)f(to)g(sho)n(w)418 4913 y(that)f(\()p Fu(H)703 4925 y Fs(0)761 4913 y Ft(\000)20 b Fu(\020)6 b Fy(\))920 4883 y Fn(\000)p Fs(1)1039 4913 y Ft(2)30 b Fu(C)1183 4925 y Fs(0)1220 4913 y Fy(\()p Fu(X)1328 4883 y Fq(])1359 4913 y Fy(\))f Ft(\032)g Fu(B)t Fy(\()p Ft(H)q Fy(\).)48 b(The)32 b(function)f Fu( )s Fy(\()p Ft(\001)p Fy(\))f(:=)f([)p Fu(h)p Fy(\()p Ft(\001)p Fy(\))21 b Ft(\000)g Fu(\020)6 b Fy(])2911 4883 y Fn(\000)p Fs(1)3032 4913 y Fy(b)r(elongs)30 b(to)418 5013 y Fu(C)477 5025 y Fs(0)515 5013 y Fy(\()p Fu(X)623 4983 y Fq(])653 5013 y Fy(\),)k(since)e Fu(h)g Fy(is)g(con)n(tin)n(uous)f(and)h(div)n(erges)e(at)i (in\014nit)n(y)h(on)f Fu(X)2620 4983 y Fq(])2650 5013 y Fy(.)51 b(Th)n(us)31 b(w)n(e)h(\014nish)h(the)418 5112 y(pro)r(of)27 b(of)h(\(a\))f(b)n(y)h(remarking)e(that)h(\()p Fu(H)1654 5124 y Fs(0)1710 5112 y Ft(\000)18 b Fu(\020)6 b Fy(\))1867 5082 y Fn(\000)p Fs(1)1980 5112 y Fy(=)23 b Fu( )s Fy(\()p Fu(P)12 b Fy(\).)p eop %%Page: 11 11 bop 1026 314 a Fw(C*{ALGEBRAS)29 b(AND)f(THE)g(ESSENTIAL)g(SPECTR)n(UM)565 b(11)518 513 y Fy(\(b\))34 b(Let)g(us)f(denote)h(b)n(y)1324 491 y Ft(\030)1324 517 y Fy(=)1422 513 y(a)f(canonical)f(isomorphism)h(and)g (b)n(y)g Fu(,)-14 b Ft(!)34 b Fy(a)f(monomorphism)418 613 y(of)d (C*-algebras.)41 b(By)30 b(applying)f(successiv)n(ely)g FA(\(E\))p Fy(,)h FA(\(D\))p Fy(,)g FA(\(B\))p Fy(,)h(Prop)r(osition)d(1.7.\(b\),)j FA(\(C\))418 712 y Fy(and)d FA(\(D\))g Fy(again,)e(w)n(e)h(ha)n(v)n(e:)637 934 y Fk(C)p Fu(=K)6 b Fy(\()p Ft(H)q Fy(\))22 b(=)1052 866 y Fg(\012)1091 934 y Ft(C)h(\001)18 b Fu(C)1258 946 y Fs(0)1296 934 y Fy(\()p Fu(X)1404 899 y Fq(])1435 934 y Fy(\))1467 866 y Fg(\013)1520 934 y Fu(=)1576 866 y Fg(\012)1615 934 y Fu(C)1674 946 y Fs(0)1711 934 y Fy(\()p Fu(X)7 b Fy(\))18 b Ft(\001)h Fu(C)1970 946 y Fs(0)2008 934 y Fy(\()p Fu(X)2116 899 y Fq(])2146 934 y Fy(\))2178 866 y Fg(\013)2241 911 y Ft(\030)2241 938 y Fy(=)2328 934 y([)p Ft(C)k Fr(o)18 b Fu(X)7 b Fy(])p Fu(=)p Fy([)p Fu(C)2724 946 y Fs(0)2761 934 y Fy(\()p Fu(X)g Fy(\))18 b Fr(o)g Fu(X)7 b Fy(])3124 911 y Ft(\030)3124 938 y Fy(=)964 1123 y Ft(\030)964 1149 y Fy(=)1052 1145 y([)p Ft(C)e Fu(=C)1225 1157 y Fs(0)1261 1145 y Fy(\()p Fu(X)i Fy(\)])19 b Fr(o)f Fu(X)29 b(,)-14 b Ft(!)1740 1003 y Fg(")1800 1066 y(Y)1788 1244 y Fq(a)p Fn(2)p Fq(A)1919 1079 y Fn(0)1942 1145 y Fu(C)6 b Fy(\()p Fu(Z)2096 1157 y Fq(a)2136 1145 y Fy(\))2168 1003 y Fg(#)2236 1145 y Fr(o)18 b Fu(X)29 b(,)-14 b Ft(!)964 1378 y Fu(,)g Ft(!)1092 1299 y Fg(Y)1079 1477 y Fq(a)p Fn(2)p Fq(A)1224 1378 y Fy([)p Fu(C)6 b Fy(\()p Fu(Z)1401 1390 y Fq(a)1441 1378 y Fy(\))19 b Fr(o)f Fu(X)7 b Fy(])1697 1355 y Ft(\030)1697 1382 y Fy(=)1797 1299 y Fg(Y)1784 1477 y Fq(a)p Fn(2)p Fq(A)1929 1310 y Fg(\012)1968 1378 y Fu(\032)2011 1390 y Fq(a)2051 1378 y Fy([)p Fu(C)f Fy(\()p Fu(Z)2228 1390 y Fq(a)2268 1378 y Fy(\)])19 b Ft(\001)g Fu(C)2443 1390 y Fs(0)2480 1378 y Fy(\()p Fu(X)2588 1343 y Fq(])2619 1378 y Fy(\))2651 1310 y Fg(\013)2704 1378 y Fu(:)518 1679 y Fy(W)-7 b(e)34 b(shall)g(denote)g(b)n(y)g Fu(I)1300 1691 y Fn(D)1392 1679 y Fy(the)h(resulting)e(monomorphism.)55 b(By)34 b(k)n(eeping)g(trac)n(k)f(of)h(the)418 1779 y(explicit)g(form)f(of)h(all)f (the)h(*-morphisms)e(w)n(e)h(see)g(that)h Fu(I)2276 1791 y Fn(D)2368 1779 y Fy(transforms)e(the)i(class)e(mo)r(dulo)418 1899 y Fu(K)6 b Fy(\()p Ft(H)q Fy(\))28 b(of)f Fu(V)19 b( )s Fy(\()p Fu(P)12 b Fy(\))28 b(in)n(to)1202 1807 y Fg(\020)1251 1899 y Fu(\032)1294 1911 y Fq(a)1348 1807 y Fg(h)1400 1878 y Fy(^)1387 1899 y Fu(V)19 b Ft(j)1477 1911 y Fq(Z)1522 1919 y Fm(a)1563 1807 y Fg(i)1616 1899 y Fu( )s Fy(\()p Fu(P)12 b Fy(\))1802 1807 y Fg(\021)1852 1957 y Fq(a)p Fn(2)p Fq(A)1987 1899 y Fy(,)28 b(whic)n(h)f(is)h(equal)f(to)2679 1831 y Fg(\000)2717 1899 y Fu(V)2765 1911 y Fq(a)2806 1899 y Fy(\()p Fu(H)2907 1911 y Fs(0)2963 1899 y Ft(\000)18 b Fu(\020)6 b Fy(\))3120 1869 y Fn(\000)p Fs(1)3210 1831 y Fg(\001)3248 1932 y Fq(a)p Fn(2)p Fq(A)3383 1899 y Fy(.)518 2038 y(Let)39 b(us)f(denote)h(b)n(y)f Fu(\031)45 b Fy(:)d Fk(C)f Ft(!)h Fk(C)p Fu(=K)6 b Fy(\()p Ft(H)q Fy(\))38 b(the)h(canonical)f(surjection.)70 b(Then)39 b Fu(\031)s Fy(\()p Fu(H)7 b Fy(\))39 b(is)418 2137 y(the)k(image)f(of)g Fu(H)49 b Fy(in)43 b(the)g(Calkin)f(algebra.)80 b(Using)42 b(\(1-2\),)k(w)n(e)c(see)g(that)h Fu(I)2986 2149 y Fn(D)3058 2137 y Fy([)p Fu(\031)s Fy(\()p Fu(H)7 b Fy(\)])48 b(=)418 2237 y(\()p Fu(H)519 2249 y Fq(a)560 2237 y Fy(\))592 2249 y Fq(a)p Fn(2)p Fq(A)727 2237 y Fy(,)d(in)e(the)f(sense)g(of)g(De\014nition)g (1.2.,)j(where)d Fu(H)2323 2249 y Fq(a)2410 2237 y Fy(=)47 b Fu(H)2591 2249 y Fs(0)2656 2237 y Fy(+)28 b Fu(V)2797 2249 y Fq(a)2879 2237 y Fy(is)42 b(a\016liated)g(to)418 2337 y Fk(C)469 2349 y Fq(a)532 2337 y Fy(:=)643 2269 y Fg(\012)682 2337 y Fu(\032)725 2349 y Fq(a)765 2337 y Fy([)p Fu(C)6 b Fy(\()p Fu(Z)942 2349 y Fq(a)983 2337 y Fy(\)])18 b Ft(\001)h Fu(C)1157 2349 y Fs(0)1195 2337 y Fy(\()p Fu(X)1303 2307 y Fq(])1333 2337 y Fy(\))1365 2269 y Fg(\013)1405 2337 y Fy(.)518 2442 y(No)n(w)i(w)n(e)g(need)g(only)g(to)h(observ)n(e)d(that)j(the)g(sp)r(ectrum)g (of)f Fu(\031)s Fy(\()p Fu(H)7 b Fy(\))22 b(is)f(the)h(essen)n(tial)f(sp)r (ectrum)418 2542 y(of)28 b Fu(H)34 b Fy(and)28 b(that)1324 2672 y Fu(\033)-51 b(\033)1317 2668 y(\033)1368 2594 y Fd(Q)1430 2656 y Fm(a)1478 2638 y Fi(C)1520 2646 y Fm(a)1571 2672 y Fy([\()p Fu(H)1695 2684 y Fq(a)1736 2672 y Fy(\))1768 2684 y Fq(a)p Fn(2)p Fq(A)1903 2672 y Fy(])23 b(=)p 2037 2575 440 4 v 2056 2593 a Fg([)2037 2771 y Fq(a)p Fn(2)p Fq(A)2179 2672 y Fu(\033)-51 b(\033)2173 2668 y(\033)2223 2648 y Fi(C)2265 2656 y Fm(a)2302 2672 y Fy(\()p Fu(H)2403 2684 y Fq(a)2444 2672 y Fy(\))p Fu(:)418 2926 y Fy(This)28 b(\014nishes)f(the)h(pro)r(of.)83 b Ff(\003)518 3097 y FA(Remark.)34 b Fy(Let)20 b(us)g(denote)g(b)n(y)g Fu(C)1583 3067 y Fq(Z)1628 3075 y Fm(a)1669 3097 y Fy(\()1725 3076 y(^)1701 3097 y Fu(X)6 b Fy(\))21 b(the)f(in)n(v)-5 b(arian)n(t)19 b(ideal)h (consisting)g(of)g(the)g(elemen)n(ts)418 3205 y(in)41 b Fu(C)6 b Fy(\()649 3184 y(^)625 3205 y Fu(X)h Fy(\))40 b(whic)n(h)h(are)e(n)n(ull)i (on)f Fu(Z)1537 3217 y Fq(a)1576 3205 y Fy(.)76 b(Another)40 b(w)n(a)n(y)f(to)h(express)f(the)i(core)e(of)i(the)g(pro)r(of)418 3329 y(of)g(the)g(p)r(oin)n(t)g(\(b\))g(is)g(to)g(sa)n(y)e(that)i(in)g(the)g (C*-algebra)2324 3237 y Fg(D)2374 3329 y Fu(C)6 b Fy(\()2496 3308 y(^)2471 3329 y Fu(X)h Fy(\))19 b Ft(\001)f Fu(C)2698 3341 y Fs(0)2736 3329 y Fy(\()p Fu(X)2844 3299 y Fq(])2875 3329 y Fy(\))2907 3237 y Fg(E)2958 3329 y Fy(,)44 b(the)d(family)418 3395 y Fg(\020D)518 3487 y Fu(C)583 3457 y Fq(Z)628 3465 y Fm(a)669 3487 y Fy(\()725 3466 y(^)701 3487 y Fu(X)7 b Fy(\))19 b Ft(\001)f Fu(C)928 3499 y Fs(0)966 3487 y Fy(\()p Fu(X)1074 3457 y Fq(])1104 3487 y Fy(\))1136 3395 y Fg(E\021)1237 3545 y Fq(a)p Fn(2)p Fq(A)1405 3487 y Fy(is)33 b(a)1569 3420 y Fg(\012)1608 3487 y Fu(C)1667 3499 y Fs(0)1705 3487 y Fy(\()p Fu(X)7 b Fy(\))18 b Ft(\001)h Fu(C)1964 3499 y Fs(0)2001 3487 y Fy(\()p Fu(X)2109 3457 y Fq(])2140 3487 y Fy(\))2172 3420 y Fg(\013)2211 3487 y Fy(-su\016cien)n(t)33 b(family)g(of)h(ideals)e(in)i(the)418 3620 y(sense)f(of)g(De\014nition)h(1.3.)52 b(Then)33 b(one)g(applies)g(Prop)r (osition)e(1.5.\(b\);)36 b(but,)f(of)e(course,)h(the)418 3720 y(explicit)28 b(form)f(of)h(the)g(observ)-5 b(ables)26 b Fu(H)1651 3732 y Fq(a)1714 3720 y Fy(=)d Fu(\031)1849 3732 y Fq(a)1889 3720 y Fy([)p Fu(H)7 b Fy(])28 b(has)f(to)g(b)r(e)h(calculated.)1084 3949 y Fz(2.)46 b(Potentials)30 b(of)g(class)h Ft(h)p Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))19 b Ft(\001)f Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\))p Ft(i)518 4104 y Fy(W)-7 b(e)35 b(shall)g(in)n(tro)r(duce)g (here)g(t)n(w)n(o)f(imp)r(ortan)n(t)h(C*-algebras:)50 b(\(a\))35 b Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))36 b(consisting)e(of)418 4204 y(con)n(tin)n(uous)j(functions)i(:)h Fu(X)47 b Ft(!)41 b Fr(C)59 b Fy(ha)n(ving)37 b(oscillation)g(whic)n(h)h(is)g(small)g(at)g (in\014nit)n(y)g(and)418 4304 y(\(b\))e Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\),)37 b(the)e(C*-algebra)d(of)j(con)n(tin)n(uous,)h(almost)e(p)r (erio)r(dic)h(functions)g(on)g Fu(X)7 b Fy(.)58 b(The)418 4409 y(sp)r(ectrum)27 b(of)g Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))27 b(is)f(the)h(largest)f(compacti\014cation)2310 4388 y(~)2287 4409 y Fu(X)33 b Fy(of)26 b Fu(X)33 b Fy(suc)n(h)27 b(that)g(the)g(action)f (of)418 4515 y(translations)j(in)i Fu(R)e Fy(:=)1206 4494 y(~)1182 4515 y Fu(X)e Ft(n)20 b Fu(X)37 b Fy(is)31 b(trivial.)46 b(The)30 b(sp)r(ectrum)h(of)g Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\))31 b(is)f(the)h(Bohr)f (group)418 4614 y Fu(T)12 b Fy(.)53 b(The)34 b(ab)r(elian)f(C*-algebra)e (generated)h(b)n(y)h Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))34 b(and)f Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\))33 b(is)h(a)f(v)n(ery)f(nice)h (one;)418 4714 y(its)25 b(sp)r(ectrum)g(is)g(the)h(union)f(b)r(et)n(w)n(een)g Fu(X)31 b Fy(and)25 b(the)g(cartesian)f(pro)r(duct)h(of)g(the)g(t)n(w)n(o)f (compact)418 4813 y(spaces)36 b Fu(R)i Fy(and)f Fu(T)12 b Fy(.)64 b(The)37 b(\014nest)g(in)n(v)-5 b(arian)n(t)36 b(decomp)r(osition)h(of)g Fu(Z)44 b Fy(:=)39 b Fu(R)25 b Ft(\002)f Fu(T)48 b Fy(is)37 b(\()p Fu(T)12 b Fy(\))3252 4825 y Fq(r)r Fn(2)p Fq(R)3384 4813 y Fy(.)418 4913 y(By)29 b(applying)f(Theorem)g(1.8.)39 b(one)28 b(gets)h(results)f(on)g(the)h(essen)n(tial)f(sp)r(ectra)g(of)h(Sc)n (hr\177)-42 b(odinger)418 5013 y(op)r(erators)28 b(with)j(p)r(oten)n(tials)e (in)h(the)h(C*-algebra)c(generated)i(b)n(y)h Fu(V)18 b(O)r Fy(\()p Fu(X)7 b Fy(\))31 b(and)f Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\).)44 b(In)418 5112 y(this)28 b(case,)f(the)h(asymptotic)f (Hamiltonians)g(are)g(all)g(almost)g(p)r(erio)r(dic.)p eop %%Page: 12 12 bop 418 314 a Fw(12)1076 b(MARIUS)28 b(M)1953 297 y(\025)1944 314 y(ANTOIU)418 513 y FA([2.1])i(The)i(C*-algebra)h(V)m(O\(X\).)518 613 y Fy(An)e(ab)r(elian,)h(lo)r(cally)e(compact,)h(non-compact)f(group)g Fu(X)37 b Fy(will)31 b(b)r(e)g(\014xed)g(from)g(no)n(w)f(on.)418 712 y(W)-7 b(e)28 b(denote)g(b)n(y)g Fu(\013)g Fy(the)h(action)e(of)h Fu(X)34 b Fy(on)28 b(itself)h(b)n(y)e(translations)g(\()p Fu(\013)2589 724 y Fq(y)2629 712 y Fy(\()p Fu(x)p Fy(\))e(=)e Fu(x)c Fy(+)f Fu(y)s Fy(\))28 b(and)g(also)418 812 y(the)g(induced)g(action)f(on)g (functions)h(\()p Fu(\013)1677 824 y Fq(y)1718 812 y Fy(\()p Fu(')p Fy(\))c(=)e Fu(')d Ft(\016)f Fu(\013)2133 824 y Fn(\000)p Fq(y)2248 812 y Fy(=)23 b Fu(')p Fy(\()p Ft(\001)c(\000)f Fu(y)s Fy(\)\).)418 956 y FA(De\014nition)40 b(2.1.)46 b Fj(W)-7 b(e)35 b(sa)n(y)d(that)j(a)e(function)h Fu(')g Ft(2)g Fu(B)t(C)2262 968 y Fq(u)2306 956 y Fy(\()p Fu(X)7 b Fy(\))34 b(has)f(asymptotic)g(v)-5 b(anishing)418 1056 y(oscillation)27 b Fj(and)g(w)n(e)g(write)h Fu(')23 b Ft(2)g Fu(V)c(O)r Fy(\()p Fu(X)7 b Fy(\))29 b Fj(if)f(for)f(an)n(y) g Fu(y)e Ft(2)f Fu(X)7 b Fj(,)27 b Fu(\013)2447 1068 y Fq(y)2487 1056 y Fy(\()p Fu(')p Fy(\))19 b Ft(\000)f Fu(')23 b Ft(2)h Fu(C)2922 1068 y Fs(0)2959 1056 y Fy(\()p Fu(X)7 b Fy(\))p Fj(.)518 1200 y Fy(Ob)n(viously)-7 b(,)30 b Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))31 b(is)g(a)g(unital)g(C*-algebra)d(placed)i(b)r(et)n(w)n(een) h Fu(C)2701 1212 y Fs(0)2739 1200 y Fy(\()p Fu(X)7 b Fy(\))30 b(and)h Fu(B)t(C)3200 1212 y Fq(u)3244 1200 y Fy(\()p Fu(X)7 b Fy(\),)418 1300 y(whic)n(h)34 b(is)f(stable)h(b)n(y)f(translations.)54 b(T)-7 b(o)33 b(get)h(a)f(link)h(with)g(a)g(more)f(traditional)g (de\014nition,)418 1400 y(let)c(us)g(set)g(for)g(an)n(y)f(function)i Fu( )e Fy(:)d Fu(X)32 b Ft(!)25 b Fr(C)50 b Fy(and)29 b(an)n(y)g Fu(K)h Ft(\032)25 b Fu(X)36 b Fy(compact)28 b(neigh)n(b)r(ourho)r(o)r(d)g(of) 418 1499 y(the)33 b(origin)e(osc)917 1511 y Fq(K)981 1499 y Fy(\()p Fu( )s Fy(\))h(:)f Fu(X)37 b Ft(!)31 b Fr(R)1462 1511 y Fs(+)1523 1499 y Fy(,)j(osc)1691 1511 y Fq(K)1755 1499 y Fy(\()p Fu( )s Fy(\)\()p Fu(x)p Fy(\))e(:=)f(sup)2263 1511 y Fq(y)r Fn(2)p Fq(K)2438 1499 y Ft(j)g Fu( )s Fy(\()p Fu(x)23 b Ft(\000)e Fu(y)s Fy(\))g Ft(\000)h Fu( )s Fy(\()p Fu(x)p Fy(\))32 b Ft(j)p Fy(.)51 b(Then)418 1599 y(it)29 b(is)g(easy)f(to)h(see)f (that)i(osc)1302 1611 y Fq(K)1395 1599 y Fy(is)e(a)h(w)n(ell-de\014ned)g(map) f(:)e Fu(B)t(C)2366 1611 y Fq(u)2409 1599 y Fy(\()p Fu(X)7 b Fy(\))26 b Ft(!)f Fu(B)t(C)2809 1611 y Fq(u)2853 1599 y Fy(\()p Fu(X)7 b Fy(\))29 b(and)f(that)h(a)418 1698 y(function)g Fu(')c Ft(2)g Fu(B)t(C)1029 1710 y Fq(u)1073 1698 y Fy(\()p Fu(X)7 b Fy(\))29 b(is)f(in)h Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))29 b(i\013)g(osc)1936 1710 y Fq(K)2000 1698 y Fy(\()p Fu(')p Fy(\))d Ft(2)f Fu(C)2283 1710 y Fs(0)2321 1698 y Fy(\()p Fu(X)7 b Fy(\).)39 b(This)29 b(sho)n(ws)e(in)i(particular)418 1798 y(that)f(if)g(one)f (de\014nes)h Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))28 b(b)n(y)f(using)g(osc) 1843 1810 y Fq(K)1907 1798 y Fy(,)g(the)h(c)n(hoice)f(of)h Fu(K)33 b Fy(is)27 b(irrelev)-5 b(an)n(t.)518 1898 y(If)29 b Fu(X)i Fy(=)25 b Fr(R)846 1868 y Fq(n)926 1898 y Fy(or)j Fu(X)j Fy(=)25 b Fr(Z)1280 1868 y Fq(n)1320 1898 y Fy(,)k Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))29 b(con)n(tains)f(the)h(C*-algebra)d Fu(C)2630 1868 y Fs(rad)2731 1898 y Fy(\()p Fu(X)7 b Fy(\))28 b(of)h(all)g(con)n(tin)n(u-)418 1997 y(ous)h(functions)h(admitting)g(radial)e (limits)i(at)f(in\014nit)n(y)-7 b(,)32 b(i.e.)46 b(those)30 b(whic)n(h)h(can)f(b)r(e)h(extended)418 2100 y(con)n(tin)n(uously)21 b(on)h(the)g(radial)f(compacti\014cation)2012 2079 y(\025)1989 2100 y Fu(X)28 b Fy(of)22 b Fu(X)28 b Fy(\(obtained)22 b(b)n(y)g(adding)f(a)h (sphere)f(at)418 2200 y(in\014nit)n(y\).)37 b(Anisotropic)25 b(Sc)n(hr\177)-42 b(odinger)25 b(op)r(erators)f(with)i(p)r(oten)n(tials)g(in) 2697 2133 y Fg(\012)2736 2200 y Fu(C)2801 2170 y Fs(rad)2901 2200 y Fy(\()p Fu(X)7 b Fy(\))18 b Ft(\001)h Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\))3368 2133 y Fg(\013)418 2300 y Fy(w)n(ere)23 b(treated)h(in)g([8])f(b)n(y)h(a)f(somewhat)h(di\013eren)n(t)g(tec)n(hnique,) g(whic)n(h)g(cannot)g(b)r(e)g(extended)g(to)418 2399 y Ft(h)p Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))19 b Ft(\001)g Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\))p Ft(i)p Fy(.)43 b(In)29 b(fact)h Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))30 b(is)f(m)n(uc)n(h)h(larger)d(than)j Fu(C)2528 2369 y Fs(rad)2628 2399 y Fy(\()p Fu(X)7 b Fy(\),)30 b(so)f(Theorem)g(2.5.)418 2499 y(will)g(extend)g(considerably)e(the)i(result) f(in)h([8].)40 b(If)29 b Fu(X)i Fy(=)24 b Fr(R)2278 2469 y Fq(n)2358 2499 y Fy(for)k(example,)g(all)h(the)g(b)r(ounded)418 2598 y Fu(C)483 2568 y Fs(1)521 2598 y Fy(-functions)f Fu(')h Fy(suc)n(h)f(that)h Ft(r)p Fu(')g Fy(is)f(of)h(class)f Fu(C)1946 2610 y Fs(0)2012 2598 y Fy(are)f(in)i Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))29 b(and)f(they)h(need)g(not)f(ha)n(v)n(e)418 2698 y(radial)j(limits.)50 b(A)32 b(simple)g(t)n(ypical)f(example)g(is)h Fu(')p Fy(\()p Fu(x)p Fy(\))g(:=)d Fu(f)23 b Fy(\()p Ft(j)p Fu(x)p Ft(j)2487 2668 y Fq(p)2526 2698 y Fy(\))32 b(for)f Fu(p)f Ft(2)g Fy([)p Ft(\0001)p Fu(;)14 b Fy(1\))32 b(and)f Fu(f)418 2798 y Fy(a)i Fu(C)558 2768 y Fs(1)630 2798 y Fy(p)r(erio)r(dic)g(function)h (of)g(one)f(v)-5 b(ariable.)54 b(In)34 b(addition,)h(the)f(fact)g(that)g Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))34 b(mak)n(es)418 2897 y(sense)27 b(for)g(an)n(y)g(ab)r(elian,)g(lo)r(cally)g(compact)g(group)g Fu(X)34 b Fy(ma)n(y)27 b(b)r(e)h(considered)e(v)n(ery)h(satisfying.)518 2998 y(Let)g(us)h(denote)g(b)n(y)1180 2977 y(~)1156 2998 y Fu(X)h Fy(=)23 b Fu(X)i Ft(t)18 b Fu(R)29 b Fy(the)f(sp)r(ectrum)f(of)h Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\).)418 3149 y FA(Lemma)39 b(2.2.)46 b Fj(\(a\))34 b(The)g(extension)g(to)1783 3128 y Fy(~)1759 3149 y Fu(X)40 b Fj(of)34 b(the)h(action)e(of)h Fu(X)40 b Fj(\(also)34 b(denoted)g(b)n(y)f Fu(\013)p Fj(\))i(is)418 3248 y(trivial)27 b(on)g Fu(R)q Fj(:)37 b(all)27 b(the)h(elemen)n(ts)g(of)f Fu(R)i Fj(are)d(\014xed)i(p)r(oin)n(ts.)518 3348 y(\(b\))g Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))28 b Fj(is)g(the)g(largest)e Fu(X)7 b Fj(-in)n(v)-5 b(arian)n(t)25 b(C*-subalgebra)g(of)j Fu(B)t(C)2668 3360 y Fq(u)2712 3348 y Fy(\()p Fu(X)7 b Fy(\))27 b Fj(satisfying)g(\(a\).)418 3492 y Fv(Pr)l(o)l(of.)44 b Fy(\(a\))20 b(Let)g(us)f(consider)g(an)g(elemen)n(t)h Fu(r)j Fy(of)c Fu(R)q Fy(,)i(i.e.)35 b(a)19 b(non-n)n(ull)g(morphism)g Fu(r)26 b Fy(:)d Fu(V)c(O)r Fy(\()p Fu(X)7 b Fy(\))24 b Ft(!)418 3592 y Fr(C)56 b Fy(whose)34 b(restriction)g(to)h Fu(C)1338 3604 y Fs(0)1375 3592 y Fy(\()p Fu(X)7 b Fy(\))35 b(is)g(n)n(ull.)58 b(F)-7 b(or)35 b(an)n(y)f Fu(y)j Ft(2)f Fu(X)41 b Fy(and)35 b Fu(')g Ft(2)g Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))36 b(w)n(e)e(ha)n(v)n(e) 418 3691 y([)p Fu(\013)494 3703 y Fq(y)534 3691 y Fy(\()p Fu(r)r Fy(\))20 b Ft(\000)e Fu(r)r Fy(][)p Fu(')p Fy(])24 b(=)f Fu(r)r Fy([)p Fu(')p Fy(\()p Ft(\001)d Fy(+)e Fu(y)s Fy(\))g Ft(\000)g Fu(')p Fy(])24 b(=)e(0,)28 b(b)n(y)f(the)h(de\014nition)g(of)f Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\).)518 3791 y(\(b\))28 b(follo)n(ws)f(b)n (y)g(a)g(sligh)n(t)g(rearrangemen)n(t)e(of)j(the)g(pro)r(of)f(of)g(the)h(p)r (oin)n(t)g(\(a\).)518 3935 y(T)-7 b(o)29 b(apply)h(Theorem)f(1.8.)44 b(to)30 b(generalized)e(Sc)n(hr\177)-42 b(odinger)29 b(op)r(erators)f(with)i (p)r(oten)n(tials)g(of)418 4035 y(class)21 b Fu(V)e(O)r Fy(\()p Fu(X)7 b Fy(\),)23 b(w)n(e)e(b)r(egin)h(b)n(y)f(noting)h(that)f(for)h(an)n(y) e(dense)i(subset)f Fu(R)2598 4047 y Fs(0)2657 4035 y Fy(of)h Fu(R)q Fy(,)h Ft(D)i Fy(:=)e(\()p Ft(f)p Fu(r)r Ft(g)p Fy(\))3243 4047 y Fq(r)r Fn(2)p Fq(R)3371 4055 y Fe(0)418 4135 y Fy(is)33 b(an)g(in)n(v)-5 b(arian)n(t)33 b(decomp)r(osition)f(of)i(the)f(dynamical)g (system)g Ft(R)g Fy(:=)f(\()p Fu(R)q(;)14 b(\013;)g(X)7 b Fy(\).)54 b(The)34 b(C*-)418 4234 y(algebra)d Fu(C)6 b Fy(\()p Ft(f)p Fu(r)r Ft(g)p Fy(\))33 b(will)g(b)r(e,)h(of)f(course,)f(iden)n(ti\014ed)h(to) g Fr(C)15 b Fy(,)40 b(in)n(terpreted)32 b(as)f(a)i(C*-subalgebra)418 4334 y(either)28 b(of)f Fu(B)t(C)874 4346 y Fq(u)918 4334 y Fy(\()p Fu(X)7 b Fy(\))28 b(or)e(of)i Fu(B)t Fy(\()p Ft(H)q Fy(\),)g Ft(H)c Fy(=)f Fu(L)1774 4304 y Fs(2)1811 4334 y Fy(\()p Fu(X)7 b Fy(\).)36 b(Th)n(us)28 b(w)n(e)f(ha)n(v)n(e)f(the)i(structure)f (result)1139 4460 y Fg(\012)1178 4527 y Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))19 b Ft(\001)f Fu(C)1569 4539 y Fs(0)1607 4527 y Fy(\()p Fu(X)1715 4493 y Fq(])1745 4527 y Fy(\))1777 4460 y Fg(\013)1831 4527 y Fu(=K)6 b Fy(\()p Ft(H)q Fy(\))22 b Fu(,)-14 b Ft(!)2249 4448 y Fg(Y)2222 4626 y Fq(r)r Fn(2)p Fq(R)2350 4634 y Fe(0)2396 4527 y Fu(C)2455 4539 y Fs(0)2493 4527 y Fy(\()p Fu(X)2601 4493 y Fq(])2631 4527 y Fy(\))p Fu(:)418 4805 y Fy(F)-7 b(or)22 b Fu(V)42 b Ft(2)23 b Fu(V)c(O)r Fy(\()p Fu(X)7 b Fy(\),)24 b(the)f(image)1434 4784 y(~)1422 4805 y Fu(V)18 b Fy(\()p Fu(R)q Fy(\))23 b(of)g Fu(R)g Fy(through)e(the)i(extension)2636 4784 y(~)2623 4805 y Fu(V)41 b Fy(of)23 b Fu(V)41 b Fy(to)3011 4784 y(~)2987 4805 y Fu(X)30 b Fy(:=)22 b Fu(X)15 b Ft(t)8 b Fu(R)418 4913 y Fy(is)23 b(just)h Fv(the)i(asymptotic)g(r)l(ange)e Fy(of)f Fu(V)c Fy(,)24 b(i.e.)35 b(the)24 b(set)f Fu(V)c Fy(\()p Fu(X)7 b Fy(\))2240 4925 y Fs(asy)2361 4913 y Fy(:=)23 b Ft(\\)2527 4928 y Fq(K)t Fn(2)p Fq(\024)p Fs(\()p Fq(X)5 b Fs(\))p 2786 4840 362 4 v 2786 4913 a Fu(V)19 b Fy(\()p Fu(X)25 b Ft(n)18 b Fu(K)6 b Fy(\))o(,)25 b(where)418 5013 y Fu(\024)p Fy(\()p Fu(X)7 b Fy(\))29 b(is)h(the)g(set)f(of)h(all)f(compact)g(neigh)n(b)r(ourho)r (o)r(ds)f(of)h(the)h(origin)f(in)g Fu(X)7 b Fy(.)42 b(Then)30 b(Theorem)418 5112 y(1.8.)36 b(giv)n(es:)p eop %%Page: 13 13 bop 1026 314 a Fw(C*{ALGEBRAS)29 b(AND)f(THE)g(ESSENTIAL)g(SPECTR)n(UM)565 b(13)418 513 y FA(Corollary)44 b(2.3.)i Fj(Let)36 b Fu(h)h Fy(:)g Fu(X)1427 483 y Fq(])1495 513 y Ft(!)g Fr(R)42 b Fj(a)36 b(con)n(tin)n(uous)f(function)h(div)n(erging)f(at)h(in\014nit)n(y)g(b)n(y)418 613 y(p)r(ositiv)n(e)k(v)-5 b(alues.)73 b(Let)40 b Fu(V)63 b Ft(2)44 b Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))p Fj(.)74 b(Then)40 b(the)h(self-adjoin)n(t)e(op)r(erator)f(acting)i(in)g Ft(H)418 712 y Fu(H)30 b Fy(:=)23 b Fu(h)p Fy(\()p Fu(P)12 b Fy(\))i(+)g Fu(V)43 b Fj(is)25 b(a\016liated)g(to)g(the)h(C*-algebra)2066 645 y Fg(\012)2106 712 y Fu(V)18 b(O)r Fy(\()p Fu(X)7 b Fy(\))19 b Ft(\001)g Fu(C)2497 724 y Fs(0)2535 712 y Fy(\()p Fu(X)2643 682 y Fq(])2673 712 y Fy(\))2705 645 y Fg(\013)2770 712 y Fj(and)25 b(for)g(an)n(y)f(dense)418 812 y(subset)k Fu(R)736 824 y Fs(0)801 812 y Fj(of)f Fu(R)i Fj(w)n(e)e(ha)n(v)n(e)418 1019 y Fy(\(2-1\))171 b Fu(\033)-51 b(\033)759 1015 y(\033)809 1031 y Fs(ess)909 1019 y Fy([)q Fu(h)p Fy(\()p Fu(P)12 b Fy(\))18 b(+)g Fu(V)h Fy(])k(=)p 1412 922 712 4 v 1446 940 a Fg([)1412 1118 y Fq(r)r Fn(2)p Fq(R)1540 1126 y Fe(0)1583 1019 y Fu(\033)-51 b(\033)1577 1015 y(\033)1628 1019 y Fy([)p Fu(h)p Fy(\()p Fu(P)12 b Fy(\))18 b(+)1942 998 y(~)1929 1019 y Fu(V)h Fy(\()p Fu(r)r Fy(\)])25 b(=)d([)28 b(min)14 b Fu(h)19 b Fy(+)f(min)c Fu(V)19 b Fy(\()p Fu(X)7 b Fy(\))2946 1031 y Fs(asy)3044 1019 y Fu(;)14 b Ft(1)p Fy(\))g Fu(:)518 1283 y Fy(It)33 b(seems)f(that)g(this)h(simple)g(result)f (do)r(es)g(not)h(app)r(ear)e(in)i(the)g(literature.)51 b(It)33 b(will)f(b)r(e)h(a)418 1383 y(rather)j(trivial)g(particular)g(case)g(of)h (our)f(Theorem)g(2.5.)65 b(W)-7 b(e)37 b(put)h(it)f(in)n(to)g(evidence)f (here)418 1482 y(b)r(ecause)26 b(it)h(has)f(a)g(feature)g(that)h(is)f(sp)r (eci\014c)g(to)h Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\):)36 b(the)27 b(essen)n(tial)f(sp)r(ectrum)g(is)h(deter-)418 1582 y(mined)j(b)n(y)f Fv(the)j(values)e Fy(tak)n(en)f(b)n(y)g(the)g(p)r(oten)n(tial)h(at)f (in\014nit)n(y)-7 b(.)43 b(Assume)29 b(for)g(simplicit)n(y)h(that)418 1682 y Fu(X)j Fy(=)27 b Fr(R)666 1651 y Fq(n)747 1682 y Fy(and)j Fu(h)p Fy(\()p Fu(p)p Fy(\))d(=)g Ft(j)p Fu(p)p Ft(j)1272 1651 y Fs(2)1309 1682 y Fy(.)44 b(The)30 b(b)r(ottom)g(of)g(the)h(essen)n(tial)e (sp)r(ectrum)h(of)g Ft(j)p Fu(P)12 b Ft(j)2993 1651 y Fs(2)3050 1682 y Fy(+)20 b Fu(V)49 b Fy(coin-)418 1781 y(cides)28 b(with)h(the)f(least) g(v)-5 b(alue)28 b(tak)n(en)g(b)n(y)f Fu(V)48 b Fy(at)28 b(in\014nit)n(y)-7 b(.)39 b Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))28 b(is)g(the)h(largest)e (C*-algebra)418 1881 y(of)h(p)r(oten)n(tials)f(for)g(whic)n(h)h(suc)n(h)f (kind)h(of)f(result)g(is)h(true)f("non-acciden)n(tally".)418 2026 y FA([2.2])j(The)i(C*-algebra)h(of)e(almost)h(p)s(erio)s(dic)h (functions)f(and)h(the)e(Bohr)h(group.)418 2172 y(De\014nition)h(2.4.)46 b Fj(A)29 b(b)r(ounded,)f(con)n(tin)n(uous)f(function)h Fu( )f Fy(:)c Fu(X)30 b Ft(!)23 b Fr(C)49 b Fj(satisfying)27 b(one)h(of)g(the)418 2272 y(next)g(equiv)-5 b(alen)n(t)27 b(conditions)g(is)h(called)f Fy(almost)g(p)r(erio)r(dic)p Fj(:)518 2371 y(\(a\))g(The)h(set)g Ft(f)p Fu( )s Fy(\()p Ft(\001)18 b Fy(+)g Fu(y)s Fy(\))23 b Ft(j)g Fu(y)j Ft(2)d Fu(X)7 b Ft(g)27 b Fj(is)g(relativ)n(ely)g(compact)g(in) h Fu(B)t(C)6 b Fy(\()p Fu(X)h Fy(\))p Fj(.)518 2471 y(\(b\))35 b(F)-7 b(or)34 b(an)n(y)g Fu(")h(>)g Fy(0)p Fj(,)h(there)f(exists)f(a)h (trigonometric)e(p)r(olynomial)h Fu(p)h Fj(on)f Fu(X)41 b Fj(suc)n(h)35 b(that)418 2571 y Ft(k)23 b Fu( )e Ft(\000)d Fu(p)23 b Ft(k)p Fu(<)f(")p Fj(.)518 2716 y Fy(W)-7 b(e)30 b(recall)f(that)i(a)e (trigonometric)g(p)r(olynomial)g(is)h(just)h(a)f(\014nite)g(linear)g(com)n (bination)f(of)418 2816 y(c)n(haracters)16 b(of)j Fu(X)7 b Fy(.)33 b(The)18 b(equiv)-5 b(alence)18 b(of)h(\(a\))f(and)g(\(b\))i(is)e(a)g (particular)f(case)g(of)i([4],)h(16.2.1.)32 b(The)418 2916 y(set)26 b(of)g(all)g(almost)g(p)r(erio)r(dic)g(functions)h(on)f Fu(X)32 b Fy(is)26 b(a)g(translational)f(in)n(v)-5 b(arian)n(t)25 b(C*-subalgebra)418 3015 y(of)37 b Fu(B)t(C)648 3027 y Fq(u)691 3015 y Fy(\()p Fu(X)7 b Fy(\),)39 b(denoted)e Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\).)64 b(All)37 b(the)f(con)n(tin)n(uous)g(functions)h(:)h Fu(X)44 b Ft(!)38 b Fr(C)58 b Fy(whic)n(h)36 b(are)418 3115 y(p)r(erio)r(dic)27 b(with)h(resp)r(ect)f(to)g(some)f(closed)h(subgroup)f (\000)h(of)g Fu(X)34 b Fy(with)27 b(compact)g(quotien)n(t)g Fu(X=)p Fy(\000)418 3214 y(lie)h(in)g Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\),)27 b(but)i(there)e(are)g(man)n(y)g(others.)518 3314 y(The)18 b(sp)r(ectrum)h(of)g Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\))19 b(is)f(a)g(compact)g(group)g Fu(T)12 b Fy(,)20 b(often)e(called)h(the)g(Bohr) e(group;)k(hence)418 3414 y(AP\(X\))33 b(is)f(canonically)f(isomorphic)g (with)h(C\(T\).)h(W)-7 b(e)32 b(recall)g(brie\015y)f(the)i(construction)e(of) 418 3513 y Fu(T)12 b Fy(;)27 b(for)g(further)g(details)h(w)n(e)f(send)h(to)f ([4])g(and)h([7].)518 3613 y(On)34 b(the)h(dual)g Fu(X)1081 3583 y Fq(])1146 3613 y Fy(of)g Fu(X)41 b Fy(w)n(e)35 b(consider)e(the)j (discrete)e(top)r(ology)-7 b(.)57 b(One)35 b(gets)f(a)g(discrete,)418 3718 y(ab)r(elian)d(group,)g(denoted)g(b)n(y)f Fu(X)1484 3678 y Fq(])1477 3743 y(d)1516 3718 y Fy(.)47 b(The)31 b(dual)g(group)f Fu(T)39 b Fy(:=)29 b(\()p Fu(X)2500 3678 y Fq(])2493 3743 y(d)2532 3718 y Fy(\))2564 3687 y Fq(])2626 3718 y Fy(with)i(the)h(usual)f(\(dual\)) 418 3817 y(top)r(ology)23 b(is)i(then)g(a)f(compact)g(group.)34 b(By)24 b(standard)g(dualit)n(y)g(argumen)n(ts)f(one)h(can)g(iden)n(tify)418 3917 y Fu(X)31 b Fy(with)26 b(a)f(dense)g(subgroup)e(of)i Fu(T)12 b Fy(.)35 b(More)25 b(precisely)-7 b(,)24 b(there)h(exists)g(a)f(con)n(tin)n (uous)g(morphism)418 4016 y Fu(\034)43 b Fy(from)32 b Fu(X)40 b Fy(to)33 b Fu(T)44 b Fy(with)33 b(dense)g(image.)52 b(W)-7 b(e)34 b(a)n(v)n(oid)d(to)i(call)f Fu(T)45 b Fy(the)33 b(Bohr)f (compacti\014cation)418 4116 y(of)j Fu(X)42 b Fy(\(as)35 b(it)h(is)f(done)g (o)r(ccasionally\),)g(since)h(the)f(top)r(ology)f(induced)i(b)n(y)f Fu(T)46 b Fy(on)35 b Fu(X)42 b Fy(\(called)418 4216 y Fv(the)31 b(Bohr)i(top)l(olo)l(gy)p Fy(\))e(is)e(usually)g(m)n(uc)n(h)g(w)n(eak)n(er)e (than)i(the)h(initial)f(one)g(\(i.e.)42 b(the)29 b(morphism)418 4315 y Fu(\034)45 b Fy(:)35 b Fu(X)42 b Ft(!)35 b Fu(T)46 b Fy(is)35 b(not)g(op)r(en\).)59 b(In)35 b(particular,)h(if)f Fu(X)41 b Fy(is)35 b(not)g(compact,)h(no)f(neigh)n(b)r(ourho)r(o)r(d)418 4415 y(of)29 b(the)h(origin)d(in)j(the)f(Bohr)f(top)r(ology)g(is)h(relativ)n (ely)f(compact)g(with)i(resp)r(ect)e(to)h(the)h(initial)418 4515 y(top)r(ology)c(on)i Fu(X)7 b Fy(.)518 4614 y(So)22 b Fu(X)29 b Fy(will)23 b(b)r(e)g(iden)n(ti\014ed)g(with)g(a)g(dense)f(subgroup) g(of)g Fu(T)12 b Fy(.)35 b(W)-7 b(e)23 b(denote)f(b)n(y)h Fu(\015)k Fy(the)c(action)f(of)418 4714 y Fu(X)33 b Fy(on)26 b Fu(T)38 b Fy(obtained)26 b(b)n(y)g(restricting)g(to)g Fu(X)d Ft(\002)16 b Fu(T)37 b Fy(the)27 b(group)f(la)n(w)g(of)g Fu(T)12 b Fy(;)26 b(it)h(extends)g(the)g(action)418 4813 y Fu(\013)e Fy(of)f Fu(X)31 b Fy(on)24 b(itself.)36 b(It)25 b(can)f(b)r(e)h(sho)n(wn)f(that)g(\() p Fu(T)7 b(;)14 b(\015)5 b(;)14 b(X)7 b Fy(\))23 b(is)i(a)f(minimal)g (dynamical)g(system,)h(i.e.)418 4913 y(the)30 b(orbit)e Ft(f)p Fu(\015)852 4925 y Fq(x)894 4913 y Fy(\()p Fu(t)p Fy(\))e Ft(j)g Fu(x)g Ft(2)g Fu(X)7 b Ft(g)28 b Fy(is)h(dense)g(in)h Fu(T)40 b Fy(for)29 b(all)g Fu(t)c Ft(2)i Fu(T)12 b Fy(.)40 b(In)30 b(fact,)g(in)f(a)g(certain)f(sense,)i(it)418 5013 y(is)f(the)g(largest)f (minimal)h(dynamical)f(system)h(with)g(group)f Fu(X)35 b Fy(for)28 b(whic)n(h)h(the)g(space)f(is)h(also)418 5112 y(a)e(group;)g(see)g([7].)p eop %%Page: 14 14 bop 418 314 a Fw(14)1076 b(MARIUS)28 b(M)1953 297 y(\025)1944 314 y(ANTOIU)418 513 y FA([2.3])i(Sc)m(hr\177)-48 b(odinger)33 b(op)s(erators)f(with)g(p)s(oten)m(tials)h(os)e(class)i Ft(h)p Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))19 b Ft(\001)f Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\))p Ft(i)q FA(.)518 613 y Fy(Let)26 b(us)g(consider)f(no)n(w)h(Sc)n(hr\177)-42 b(odinger)24 b(op)r(erators)g (with)j(p)r(oten)n(tials)f(in)g(the)h(C*-subalgebra)418 713 y Ft(A)34 b Fy(of)f Fu(B)t(C)744 725 y Fq(u)788 713 y Fy(\()p Fu(X)7 b Fy(\))34 b(generated)e(b)n(y)h Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))34 b(and)f Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\).)54 b(W)-7 b(e)34 b(shall)f(need)h(some)f(notations.)418 813 y(Let)26 b(us)f(\014x)h(an)f(elemen)n(t)g Fu(V)45 b Fy(of)25 b Ft(A)p Fy(.)37 b(It)26 b(ma)n(y)e(b)r(e)i(writen)g(as)f Fu(V)42 b Fy(=)22 b(lim)2538 825 y Fq(n)p Fn(!1)2730 750 y Fg(P)2817 771 y Fq(m)2876 779 y Fm(n)2817 838 y Fq(j)s Fs(=1)2950 813 y Fu(U)3016 782 y Fq(n)3007 834 y(j)3061 813 y Fu(W)3151 782 y Fq(n)3139 834 y(j)3196 813 y Fy(,)k(with)418 919 y Fu(m)491 931 y Fq(n)559 919 y Ft(2)e Fr(N)t Fy(,)33 b Fu(U)814 889 y Fq(n)805 941 y(j)882 919 y Ft(2)24 b Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))28 b(and)f Fu(W)1512 889 y Fq(n)1500 941 y(j)1580 919 y Ft(2)d Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\).)37 b(F)-7 b(or)27 b(an)n(y)g Fu(r)e Ft(2)f Fu(R)k Fy(and)g Fu(x)23 b Ft(2)h Fu(X)33 b Fy(w)n(e)28 b(set)418 1201 y(\(2-2\))690 b Fu(V)1332 1213 y Fq(r)1369 1201 y Fy(\()p Fu(x)p Fy(\))24 b(:=)f(lim)1730 1213 y Fq(n)p Fn(!1)1932 1096 y Fq(m)1991 1104 y Fm(n)1922 1122 y Fg(X)1924 1299 y Fq(j)s Fs(=1)2070 1180 y Fy(~)2055 1201 y Fu(U)2121 1166 y Fq(n)2112 1221 y(j)2166 1201 y Fy(\()p Fu(r)r Fy(\))p Fu(W)2359 1166 y Fq(n)2347 1221 y(j)2406 1201 y Fy(\()p Fu(x)p Fy(\))p Fu(;)418 1505 y Fy(where)676 1484 y(~)662 1505 y Fu(U)728 1475 y Fq(n)719 1527 y(j)804 1505 y Fy(is)32 b(the)f(con)n(tin)n(uous)g(extension)g(of)g Fu(U)1993 1475 y Fq(n)1984 1527 y(j)2069 1505 y Fy(to)2198 1484 y(~)2174 1505 y Fu(X)36 b Fy(=)29 b Fu(X)e Ft(t)22 b Fu(R)q Fy(.)47 b(It)32 b(will)g(b)r(ecome)f(clear)418 1605 y(in)26 b(the)h(pro)r(of)e(of)h (Theorem)f(2.5.)35 b(that)27 b(the)f(almost)f(p)r(erio)r(dic)h(function)h Fu(V)2752 1617 y Fq(r)2815 1605 y Fy(do)r(es)e(not)h(dep)r(end)418 1704 y(on)h(the)h(represen)n(tation)e(c)n(ho)r(osen)h(for)g Fu(V)19 b Fy(.)418 1856 y FA(Theorem)41 b(2.5.)46 b Fj(Let)37 b Fu(h)g Fy(:)g Fu(X)1408 1826 y Fq(])1475 1856 y Ft(!)g Fr(R)42 b Fj(con)n(tin)n(uous,)c(suc)n(h)d(that)i Fy(lim)2640 1868 y Fq(p)p Fn(!1)2825 1856 y Fu(h)p Fy(\()p Fu(p)p Fy(\))g(=)g Ft(1)p Fj(.)62 b(Let)418 1956 y Fu(V)42 b Ft(2)23 b(A)p Fj(.)518 2056 y(\(a\))31 b(The)h(self-adjoin)n(t)f(op)r(erator)f Fu(H)36 b Fy(:=)29 b Fu(h)p Fy(\()p Fu(P)12 b Fy(\))22 b(+)e Fu(V)51 b Fj(is)31 b(a\016liated)g(to)h(the)g(C*-subalgebra)418 2088 y Fg(\012)457 2156 y Ft(A)19 b(\001)g Fu(C)643 2168 y Fs(0)680 2156 y Fy(\()p Fu(X)788 2126 y Fq(])819 2156 y Fy(\))851 2088 y Fg(\013)918 2156 y Fj(of)27 b Fu(B)t Fy(\()p Ft(H)q Fy(\))p Fj(,)i Ft(H)24 b Fy(=)e Fu(L)1504 2126 y Fs(2)1541 2156 y Fy(\()p Fu(X)7 b Fy(\))p Fj(.)518 2256 y(\(b\))28 b(W)-7 b(e)28 b(denote)g(b)n(y)f Fu(H)1251 2268 y Fq(r)1315 2256 y Fj(the)h(almost)f(p)r(erio)r(dic)h(op)r (erator)e Fu(h)p Fy(\()p Fu(P)12 b Fy(\))18 b(+)g Fu(V)2706 2268 y Fq(r)2744 2256 y Fj(.)37 b(Then)1528 2491 y Fu(\033)-51 b(\033)1521 2487 y(\033)1572 2503 y Fs(ess)1658 2491 y Fy(\()p Fu(H)7 b Fy(\))24 b(=)p 1909 2394 387 4 v 1943 2412 a Fg([)1909 2590 y Fq(r)r Fn(2)p Fq(R)2037 2598 y Fe(0)2081 2491 y Fu(\033)-51 b(\033)2074 2487 y(\033)2125 2491 y Fy(\()p Fu(H)2226 2503 y Fq(r)2263 2491 y Fy(\))418 2775 y Fj(for)27 b(an)n(y)g(dense)g(subset)h Fu(R)1246 2787 y Fs(0)1311 2775 y Fj(of)g Fu(R)q Fj(.)518 2927 y FA(Example.)61 b Fy(Let)36 b(us)g(consider)f(the)h(v)n(ery)f(simple)h (example)g Fu(V)55 b Fy(=)37 b Fu(U)9 b(W)j Fy(,)37 b(where)f Fu(U)45 b Fy(has)418 3026 y(v)-5 b(anishing)25 b(oscillation)f(at)g (in\014nit)n(y)i(and)f Fu(W)37 b Fy(is)24 b(almost)h(p)r(erio)r(dic.)36 b(By)24 b(using)h(the)g(asymptotic)418 3126 y(range)h(of)i Fu(U)36 b Fy(\(in)n(tro)r(duced)28 b(in)g FA([2.1])p Fy(\))e(one)h(ma)n(y)g (write)1143 3361 y Fu(\033)-51 b(\033)1137 3357 y(\033)1187 3373 y Fs(ess)1287 3361 y Fy([)q Fu(h)p Fy(\()p Fu(P)12 b Fy(\))18 b(+)g Fu(V)h Fy(])k(=)p 1790 3264 867 4 v 1910 3282 a Fg([)1790 3464 y Fq(\026)p Fn(2)p Fq(U)6 b Fs(\()p Fq(X)f Fs(\))2037 3472 y Fe(asy)2134 3361 y Fu(\033)-51 b(\033)2128 3357 y(\033)2192 3361 y Fy([)p Fu(h)p Fy(\()p Fu(P)12 b Fy(\))19 b(+)f Fu(\026W)12 b Fy(])p Fu(:)418 3657 y Fy(The)27 b(v)-5 b(alues)26 b(tak)n(en)g(b)n(y)g Fu(U)35 b Fy(at)27 b(in\014nit)n(y)g(serv)n(e)e(as)h("coupling)f(constan)n (ts")h(for)g(the)g(asymptotic)418 3756 y(Hamiltonians)h(assigned)g(to)g Fu(H)7 b Fy(.)418 3908 y Fv(Pr)l(o)l(of)27 b(of)f(The)l(or)l(em)h(2.5.)44 b Ft(A)24 b Fy(is)f(indeed)h(a)e Fu(X)7 b Fy(-in)n(v)-5 b(arian)n(t,)22 b(unital)i(C*-subalgebra)c(of)j Fu(B)t(C)3204 3920 y Fq(u)3248 3908 y Fy(\()p Ft(H)q Fy(\).)418 4008 y(\(a\))e(follo)n(ws)f(directly)g(from) g(the)i(p)r(oin)n(t)e(\(a\))h(of)g(Theorem)f(1.8.)34 b(W)-7 b(e)21 b(shall)f(v)n(erify)g(that)h(Theorem)418 4107 y(1.8.\(b\))28 b(implies)f(our)g(p)r(oin)n(t)h(\(b\).)518 4216 y(Let)j(us)f(denote)h(b)n(y) 1193 4195 y(^)1169 4216 y Fu(X)j Fy(=)28 b Fu(X)f Ft(t)21 b Fu(Z)37 b Fy(the)31 b(sp)r(ectrum)g(of)g Ft(A)p Fy(;)i(it)e(is)g(a)f (compacti\014cation)g(of)h Fu(X)418 4315 y Fy(and)c(the)h(space)f(of)g(a)g (dynamical)g(system)g(with)h(group)e Fu(X)7 b Fy(.)36 b(It)28 b(will)f(b)r(e)h(useful)g(to)f(kno)n(w)g(that)418 4415 y(the)32 b(compact)f(space)f Fu(Z)37 b Fy(is)32 b(naturally)e(homeomorphic)g(to)i(the) f(cartesian)f(pro)r(duct)i Fu(R)21 b Ft(\002)g Fu(T)12 b Fy(.)418 4515 y(This)24 b(is)g(b)n(y)g(no)f(means)h(a)f(general)g(result.)35 b(It)25 b(expresses)d(the)i(fact)g(that)h Fu(V)19 b(O)r Fy(\()p Fu(X)7 b Fy(\))24 b(and)g Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\))418 4614 y(are)31 b("asymptotically)f(indep)r(enden)n(t",)j(a)e(concept)g(in)n (tro)r(duced)h(and)f(studied)h(b)n(y)g(P)n(o)n(w)n(er)d(in)418 4714 y(a)d(series)f(of)g(articles)g(ev)n(en)h(for)f(non-comm)n(uting)g (C*-algebras.)34 b(The)26 b(natural)f(candidate)h(for)418 4813 y(the)36 b(homeomorphism)f(is)h Fu(\022)j Fy(:)d Fu(Z)43 b Ft(!)37 b Fu(R)24 b Ft(\002)g Fu(T)47 b Fy(giv)n(en)35 b(b)n(y)g Fu(\022)r Fy(\()p Fu(z)t Fy(\))i(:=)2579 4746 y Fg(\000)2617 4813 y Fu(z)t Ft(j)2683 4828 y Fq(V)14 b(O)r Fs(\()p Fq(X)5 b Fs(\))2904 4813 y Fu(;)14 b(z)t Ft(j)3007 4828 y Fq(AP)9 b Fs(\()p Fq(X)c Fs(\))3222 4746 y Fg(\001)3260 4813 y Fy(.)62 b(It)418 4913 y(is)34 b(clear)f(that)i(it)g(is)f(a)g(w)n(ell-de\014ned)g(con) n(tin)n(uous,)h(op)r(en)f(injection.)57 b(The)34 b(non-trivial)f(part)418 5013 y(of)f(the)g(pro)r(of)e(is)i(to)f(c)n(hec)n(k)g(surjectivit)n(y)-7 b(.)48 b(But)32 b(this)g(can)f(b)r(e)h(done)f(b)n(y)h(p)r(erforming)e(ob)n (vious)418 5112 y(mo)r(di\014cations)d(in)h(the)g(pro)r(of)f(of)h(Lemma)f(1.) 36 b(from)28 b([10].)p eop %%Page: 15 15 bop 1026 314 a Fw(C*{ALGEBRAS)29 b(AND)f(THE)g(ESSENTIAL)g(SPECTR)n(UM)565 b(15)518 513 y Fy(It)33 b(is)h(clear)e(that)i Fu(\022)i Fy(is)d(ev)n(en)g(an) g(isomorphism)f(of)h Fu(X)7 b Fy(-dynamical)32 b(systems.)54 b(On)33 b Fu(R)23 b Ft(\002)f Fu(T)418 613 y Fy(w)n(e)40 b(consider)g(the)h (in)n(v)-5 b(arian)n(t)40 b(decomp)r(osition)g Ft(D)2034 582 y Fn(0)2102 613 y Fy(:=)45 b(\()p Ft(f)p Fu(r)r Ft(g)27 b(\002)g Fu(T)12 b Fy(\))2602 625 y Fq(r)r Fn(2)p Fq(R)2730 633 y Fe(0)2765 613 y Fy(;)48 b(w)n(e)40 b(transp)r(ort)g(it)418 712 y(through)30 b Fu(\022)776 682 y Fn(\000)p Fs(1)896 712 y Fy(and)g(obtain)g(an)g(in)n(v)-5 b(arian)n(t)30 b(decomp)r(osition)f Ft(D)k Fy(of)e Fu(Z)36 b Fy(consisting)29 b(of)i(minimal)418 812 y(comp)r(onen)n(ts)g Fu(Z)936 824 y Fq(r)972 812 y Fy(,)h Fu(r)f Ft(2)f Fu(R)1243 824 y Fs(0)1280 812 y Fy(.)47 b(The)31 b(C*-algebras)d Fu(C)6 b Fy(\()p Fu(Z)2134 824 y Fq(r)2171 812 y Fy(\))32 b(and)e Fu(C)6 b Fy(\()p Ft(f)p Fu(r)r Ft(g)21 b(\002)g Fu(T)12 b Fy(\))30 b(are)g(isomorphic.)418 911 y(By)d(comp)r(osing)f(with)h(the)g(natural)f (iden)n(ti\014cation)h(b)r(et)n(w)n(een)g Fu(C)6 b Fy(\()p Ft(f)p Fu(r)r Ft(g)18 b(\002)e Fu(T)c Fy(\))26 b(and)h Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\),)27 b(w)n(e)418 1019 y(get)h(an)f(isomorphism)g Fu(\032)1201 1031 y Fq(r)1261 1019 y Fy(:)c Fu(C)6 b Fy(\()p Fu(Z)1461 1031 y Fq(r)1498 1019 y Fy(\))24 b Ft(!)f Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\).)37 b(It)29 b(is)e(ob)n(vious)g(that)h Fu(\032)2685 1031 y Fq(r)2721 1019 y Fy([)2757 998 y(^)2744 1019 y Fu(V)19 b Ft(j)2834 1031 y Fq(Z)2879 1039 y Fm(r)2917 1019 y Fy(])28 b(\(notation)g(as)418 1119 y(in)23 b(Theorem)e(1.8.\))34 b(coincides)22 b(with)h Fu(V)1629 1131 y Fq(r)1688 1119 y Fy(giv)n(en)f(b)n (y)g(\(2-2\);)h(this)g(also)e(sho)n(ws)g(the)h(indep)r(endence)418 1218 y(of)28 b Fu(V)561 1230 y Fq(r)626 1218 y Fy(of)f(the)h(represen)n (tation)e(c)n(hosen)h(for)g Fu(V)19 b Fy(.)518 1318 y(No)n(w)27 b(the)h(p)r(oin)n(t)g(\(b\))g(of)g(Theorem)e(1.8.)36 b(can)28 b(b)r(e)g(applied)f(straigh)n(tforw)n(ardly)-7 b(.)80 b Ff(\003)518 1458 y FA(Remark.)38 b Fy(The)28 b(basic)g(structure)f(result)h(whic)n(h)g (allo)n(ws)f(deducing)h(Theorem)g(2.5.)37 b(from)418 1557 y(Theorem)30 b(1.8.)45 b(is)30 b(the)h(fact)g(that)g(the)f(C*-algebra)2097 1490 y Fg(\012)2137 1557 y Ft(A)18 b(\001)h Fu(C)2322 1569 y Fs(0)2359 1557 y Fy(\()p Fu(X)2467 1527 y Fq(])2498 1557 y Fy(\))2530 1490 y Fg(\013)2600 1557 y Fy(can)30 b(b)r(e)h(em)n(b)r(edded)g (in)n(to)418 1603 y Fg(Q)497 1690 y Fq(r)r Fn(2)p Fq(R)625 1698 y Fe(0)674 1598 y Fg(\012)713 1665 y Fu(AP)12 b Fy(\()p Fu(X)7 b Fy(\))19 b Ft(\001)f Fu(C)1099 1677 y Fs(0)1137 1665 y Fy(\()p Fu(X)1245 1635 y Fq(])1276 1665 y Fy(\))1308 1598 y Fg(\013)1347 1665 y Fy(.)35 b(In)21 b(turn,)h(this)f(result)g(follo)n(ws)f (from)g(prop)r(erties)g(of)h(the)g(crossed)418 1769 y(pro)r(duct,)27 b(from)e(the)i(iden)n(ti\014cation)f(of)g(the)g(sp)r(ectrum)g(of)g Ft(A)p Fu(=C)2442 1781 y Fs(0)2480 1769 y Fy(\()p Fu(X)7 b Fy(\))26 b(with)g Fu(R)16 b Ft(\002)f Fu(T)37 b Fy(and)26 b(from)418 1869 y(the)31 b(w)n(a)n(y)e(the)i(group)e Fu(X)37 b Fy(acts)30 b(on)g Fu(R)i Fy(and)e Fu(T)41 b Fy(\(trivially)31 b(on)f Fu(R)h Fy(and)f(minimally)h(on)f Fu(T)12 b Fy(\).)45 b(The)418 1968 y(p)r(oin)n(ts)26 b(of)f(the)h(space)f Fu(V)19 b Fy(\()p Fu(X)7 b Fy(\))1326 1980 y Fs(asy)1450 1968 y Fv(do)29 b(not)d Fy(serv)n(e)e(to)i (de\014ne)f(asymptotic)h(Hamiltonians)f(for)g Fu(H)7 b Fy(.)1435 2148 y Fz(3.)45 b(Car)-6 b(tesian)31 b(pr)n(oducts)518 2297 y Fy(W)-7 b(e)24 b(shall)g(study)h(here)f(pro)r(ducts)g(of)g(dynamical)g (systems.)35 b(F)-7 b(or)24 b(notational)f(con)n(v)n(enience,)418 2397 y(only)f(t)n(w)n(o)g(factors)f(will)i(b)r(e)f(considered;)h(the)g(case)f (of)g(an)g(arbitrary)-7 b(,)22 b(\014nite)h(n)n(um)n(b)r(er)f(of)g(factors) 418 2496 y(is)28 b(essen)n(tially)e(the)i(same.)518 2605 y(So,)41 b(let)d Ft(X)871 2575 y Fq(j)947 2605 y Fy(:=)j(\()1143 2578 y(^)1108 2605 y Fu(X)1184 2581 y Fq(j)1219 2605 y Fu(;)14 b(\013)1309 2575 y Fq(j)1344 2605 y Fu(;)g(X)1457 2575 y Fq(j)1491 2605 y Fy(\),)42 b Fu(j)j Fy(=)c(1)p Fu(;)14 b Fy(2)76 b(b)r(e)38 b(t)n(w)n(o)g(top)r(ological)f(dynamical)g(systems,)418 2719 y(where)696 2692 y(^)661 2719 y Fu(X)737 2695 y Fq(j)802 2719 y Fy(is)31 b(some)f(compacti\014cation)f(of)i(the)g(lo)r(cally)f(compact)g (group)f Fu(X)2884 2688 y Fq(j)2949 2719 y Fy(and)i Fu(\013)3167 2688 y Fq(j)3233 2719 y Fy(is)f(an)418 2818 y(extension)25 b(of)g(the)g(usual)g(action)g(of)g Fu(X)1645 2788 y Fq(j)1704 2818 y Fy(on)g(itself)h(b)n(y)f(translations.)34 b(W)-7 b(e)26 b(set)f Fu(X)k Fy(:=)23 b Fu(X)3166 2788 y Fs(1)3216 2818 y Ft(\002)14 b Fu(X)3371 2788 y Fs(2)418 2926 y Fy(\(direct)21 b(pro)r(duct)g(of)f(top)r(ological)g(groups\),)1828 2905 y(^)1805 2926 y Fu(X)29 b Fy(:=)2049 2900 y(^)2014 2926 y Fu(X)2090 2902 y Fs(1)2131 2926 y Ft(\002)2236 2900 y Fy(^)2201 2926 y Fu(X)2277 2902 y Fs(2)2333 2926 y Fy(\(cartesian)20 b(pro)r(duct)h(of)f (compact)418 3026 y(spaces\))35 b(and)g Fu(\013)h Fy(:=)g Fu(\013)1148 2996 y Fs(1)1209 3026 y Ft(\002)24 b Fu(\013)1351 2996 y Fs(2)1424 3026 y Fy(\(the)36 b(pro)r(duct)f(action\).)60 b(W)-7 b(e)36 b(will)f(b)r(e)h(concerned)f(with)h(the)418 3133 y(pro)r(duct)28 b(dynamical)f(system)g Ft(X)36 b Fy(:=)23 b(\()1666 3112 y(^)1642 3133 y Fu(X)6 b(;)14 b(\013;)g(X)7 b Fy(\),)28 b(so)f(the)h(ab)r(elian)g (C*-algebra)d(of)i(p)r(oten)n(tials)418 3241 y(will)j(b)r(e)g(a)g(tensor)e (pro)r(duct.)44 b(The)29 b(b)r(oundary)g Fu(Z)k Fy(:=)26 b(\()2174 3220 y(^)2150 3241 y Fu(X)2226 3211 y Fs(1)2283 3241 y Ft(\002)2391 3220 y Fy(^)2367 3241 y Fu(X)2443 3211 y Fs(2)2480 3241 y Fy(\))20 b Ft(n)f Fy(\()p Fu(X)2701 3211 y Fs(1)2758 3241 y Ft(\002)g Fu(X)2918 3211 y Fs(2)2955 3241 y Fy(\))30 b(admits)g(the)418 3354 y(ob)n(vious)d(partition)h Fu(Z)j Fy(=)24 b(\()p Fu(X)1351 3324 y Fs(1)1407 3354 y Ft(\002)18 b Fu(Z)1553 3324 y Fs(2)1590 3354 y Fy(\))i Ft(t)f Fy(\()p Fu(Z)1811 3324 y Fs(1)1867 3354 y Ft(\002)g Fu(X)2027 3324 y Fs(2)2063 3354 y Fy(\))h Ft(t)f Fy(\()p Fu(Z)2284 3324 y Fs(1)2340 3354 y Ft(\002)g Fu(Z)2487 3324 y Fs(2)2524 3354 y Fy(\),)29 b(where)f Fu(Z)2912 3324 y Fq(j)2971 3354 y Fy(:=)3118 3328 y(^)3084 3354 y Fu(X)3160 3330 y Fq(j)3213 3354 y Ft(n)19 b Fu(X)3350 3324 y Fq(j)3384 3354 y Fy(.)418 3454 y(But)33 b(note)g(that)g(among)f(the)i(three)e(in)n(v)-5 b(arian)n(t)32 b(comp)r(onen)n(ts)h(of)f(the)i(righ)n(t-hand)e(side)g(only) 418 3554 y(the)39 b(last)e(one)h(is)g(closed.)68 b(This)38 b(time,)j(in)n(v)-5 b(arian)n(t)37 b(decomp)r(ositions)g(are)g(somewhat)h (more)418 3653 y(complicated)32 b(than)g(in)g(Section)g(2;)i(they)f(do)e(not) i(form)e(partitions)h(of)g Fu(Z)6 b Fy(.)50 b(W)-7 b(e)32 b(shall)g(study)418 3753 y(some)f(of)h(them)h(and)e(write)h(corresp)r(onding)e(form)n(ulae)h(for) g(the)i(essen)n(tial)e(sp)r(ectrum)h(of)g(the)418 3853 y(a\016liated)27 b(Hamiltonians)h(in)f(an)h(in)n(teresting)f(particular)f(case.)418 3992 y FA([3.1])k(Cartesian)j(in)m(v)-5 b(arian)m(t)35 b(decomp)s(ositions.) 518 4092 y Fy(W)-7 b(e)36 b(are)e(in)n(terested)i(only)f(in)h(certain)f(in)n (v)-5 b(arian)n(t)34 b(decomp)r(ositions)h(of)h Ft(Z)43 b Fy(:=)36 b(\()p Fu(Z)q(;)14 b(\013;)g(X)7 b Fy(\),)418 4191 y(whose)27 b(elemen)n(ts)g(are)g(cartesian)f(pro)r(ducts.)418 4331 y FA(De\014nition)48 b(3.1.)f Fj(The)41 b(in)n(v)-5 b(arian)n(t)40 b(decomp)r(osition)g Ft(D)48 b Fy(:=)d(\()p Fu(Z)2516 4343 y Fq(a)2556 4331 y Fy(\))2588 4343 y Fq(a)p Fn(2)p Fq(A)2769 4331 y Ft(2)h Fr(D)14 b Fy(\()p Ft(Z)7 b Fy(\))47 b Fj(is)41 b(called)418 4438 y Fy(cartesian)29 b Fj(if)h(there)f(are)g Ft(D)1271 4408 y Fq(j)1333 4438 y Fy(:=)e(\()p Fu(Z)1543 4398 y Fq(j)1537 4467 y(a)1573 4451 y Fm(j)1608 4438 y Fy(\))1640 4454 y Fq(a)1676 4437 y Fm(j)1707 4454 y Fn(2)p Fq(A)1802 4437 y Fm(j)1864 4438 y Ft(2)g Fr(D)14 b Fy(\()p Ft(Z)2099 4408 y Fq(j)2140 4438 y Fy(\))p Fj(,)30 b Fu(j)i Fy(=)26 b(1)p Fu(;)14 b Fy(2)29 b Fj(suc)n(h)h(that)g(for)f(an)n(y)g Fu(a)e Ft(2)g Fu(A)418 4559 y Fj(either)h Fu(Z)711 4571 y Fq(a)774 4559 y Fy(=)897 4533 y(^)861 4559 y Fu(X)937 4535 y Fs(1)992 4559 y Ft(\002)19 b Fu(Z)1139 4529 y Fs(2)1133 4585 y Fq(a)1169 4569 y Fe(2)1232 4559 y Fj(for)28 b(some)f Fu(a)1612 4529 y Fs(2)1672 4559 y Ft(2)d Fu(A)1813 4529 y Fs(2)1878 4559 y Fj(or)j Fu(Z)2037 4571 y Fq(a)2100 4559 y Fy(=)22 b Fu(Z)2250 4529 y Fs(1)2244 4585 y Fq(a)2280 4569 y Fe(1)2335 4559 y Ft(\002)2454 4533 y Fy(^)2418 4559 y Fu(X)2494 4535 y Fs(2)2558 4559 y Fj(for)28 b(some)f Fu(a)2938 4529 y Fs(1)2998 4559 y Ft(2)c Fu(A)3138 4529 y Fs(1)3204 4559 y Fj(.)37 b(The)418 4659 y(set)28 b(of)f(all)h (cartesian)e(in)n(v)-5 b(arian)n(t)26 b(decomp)r(ositions)h(of)h Ft(Z)34 b Fj(will)28 b(b)r(e)g(denoted)f(b)n(y)h Fr(D)2979 4671 y Fq(c)3019 4659 y Fy(\()p Ft(Z)7 b Fy(\))p Fj(.)518 4798 y Fy(It)22 b(is)g(ob)n(vious)e(that)j(there)e(is)h(a)g(bijection)g(b)r(et)n (w)n(een)g Fr(D)2199 4810 y Fq(c)2239 4798 y Fy(\()p Ft(Z)7 b Fy(\))22 b(and)g Fr(D)14 b Fy(\()p Ft(Z)2701 4768 y Fs(1)2744 4798 y Fy(\))7 b Ft(\002)g Fr(D)14 b Fy(\()p Ft(Z)3008 4768 y Fs(2)3051 4798 y Fy(\).)36 b(Indeed,)418 4898 y(the)28 b(most)f(general)g (cartesian)f(in)n(v)-5 b(arian)n(t)26 b(decomp)r(osition)h(of)h Ft(Z)34 b Fy(has)27 b(the)h(form)418 5073 y(\(3-1\))543 b Ft(D)25 b Fy(:=)1337 4981 y Fg(\020)1422 5047 y Fy(^)1386 5073 y Fu(X)1462 5049 y Fs(1)1517 5073 y Ft(\002)18 b Fu(Z)1663 5039 y Fs(2)1657 5095 y Fq(a)1693 5079 y Fe(2)1757 5073 y Fu(;)42 b(Z)1885 5039 y Fs(1)1879 5095 y Fq(a)1915 5079 y Fe(1)1969 5073 y Ft(\002)2088 5047 y Fy(^)2052 5073 y Fu(X)2128 5049 y Fs(2)2165 4981 y Fg(\021)2215 5131 y Fs(\()p Fq(a)2277 5114 y Fe(1)2309 5131 y Fq(;a)2365 5114 y Fe(2)2397 5131 y Fs(\))p Fn(2)p Fq(A)2518 5114 y Fe(1)2550 5131 y Fn(\002)p Fq(A)2652 5114 y Fe(2)p eop %%Page: 16 16 bop 418 314 a Fw(16)1076 b(MARIUS)28 b(M)1953 297 y(\025)1944 314 y(ANTOIU)418 513 y Fy(for)f(some)g Ft(D)819 483 y Fq(j)878 513 y Fy(=)22 b(\()p Fu(Z)1060 473 y Fq(j)1054 542 y(a)1090 525 y Fm(j)1125 513 y Fy(\))1157 529 y Fq(a)1193 512 y Fm(j)1225 529 y Fn(2)p Fq(A)1320 512 y Fm(j)1378 513 y Ft(2)h Fr(D)14 b Fy(\()p Ft(Z)1609 483 y Fq(j)1650 513 y Fy(\),)28 b Fu(j)g Fy(=)23 b(1)p Fu(;)14 b Fy(2.)518 618 y(So)38 b(let)h(us)g(\014x)g(no)n(w)f Ft(D)1272 588 y Fs(1)1352 618 y Ft(2)k Fr(D)14 b Fy(\()p Ft(Z)1602 588 y Fs(1)1645 618 y Fy(\))39 b(and)g Ft(D)1955 588 y Fs(2)2034 618 y Ft(2)j Fr(D)14 b Fy(\()p Ft(Z)2285 588 y Fs(2)2327 618 y Fy(\))40 b(and)e(denote)h(b)n(y)g Ft(D)44 b(2)e Fr(D)3236 630 y Fq(c)3276 618 y Fy(\()p Ft(Z)7 b Fy(\))418 717 y(the)37 b(corresp)r(onding)e(cartesian)h(in)n(v)-5 b(arian)n(t)35 b(decomp)r(osition) h(of)h Ft(Z)7 b Fy(,)39 b(as)d(in)h(\(3-1\).)64 b(A)37 b(direct)418 817 y(application)27 b(of)h(Prop)r(osition)e(1.7.)36 b(giv)n(es)418 1078 y(\(3-2\))73 b Fu(C)6 b Fy(\()788 1057 y(^)764 1078 y Fu(X)h Fy(\))p Fu(=C)973 1090 y Fs(0)1010 1078 y Fy(\()p Fu(X)g Fy(\))23 b Fu(,)-14 b Ft(!)1289 936 y Fg( )1399 999 y(Y)1354 1180 y Fq(a)1390 1164 y Fe(2)1423 1180 y Fn(2)p Fq(A)1518 1164 y Fe(2)1550 1011 y Fn(0)1573 1078 y Fu(C)6 b Fy(\()1706 1052 y(^)1670 1078 y Fu(X)1746 1054 y Fs(1)1783 1078 y Fy(\))19 b Ft(\012)f Fu(C)6 b Fy(\()p Fu(Z)2077 1043 y Fs(2)2071 1100 y Fq(a)2107 1083 y Fe(2)2144 1078 y Fy(\))2176 936 y Fg(!)2260 1078 y Ft(\002)2343 936 y Fg( )2454 999 y(Y)2409 1180 y Fq(a)2445 1164 y Fe(1)2477 1180 y Fn(2)p Fq(A)2572 1164 y Fe(1)2604 1011 y Fn(0)2628 1078 y Fu(C)g Fy(\()p Fu(Z)2788 1043 y Fs(1)2782 1100 y Fq(a)2818 1083 y Fe(1)2854 1078 y Fy(\))19 b Ft(\012)f Fu(C)6 b Fy(\()3121 1052 y(^)3085 1078 y Fu(X)3161 1054 y Fs(2)3198 1078 y Fy(\))3230 936 y Fg(!)3310 1078 y Fu(:)518 1377 y Fy(Let)32 b(us)g(set)g Fk(C)e Fy(:=)g Fu(C)6 b Fy(\()1237 1356 y(^)1212 1377 y Fu(X)h Fy(\))22 b Fr(o)f Fu(X)7 b Fy(,)32 b Fk(K)f Fy(:=)f Fu(C)1822 1389 y Fs(0)1860 1377 y Fy(\()p Fu(X)7 b Fy(\))21 b Fr(o)g Fu(X)39 b Fy(and)31 b Fk(C)2431 1346 y Fq(j)2497 1377 y Fy(:=)f Fu(C)6 b Fy(\()2747 1350 y(^)2712 1377 y Fu(X)2788 1353 y Fq(j)2823 1377 y Fy(\))21 b Fr(o)g Fu(X)3038 1346 y Fq(j)3073 1377 y Fy(;)34 b(one)e(has)418 1476 y Fk(C)23 b Fy(=)g Fk(C)631 1446 y Fs(1)682 1476 y Ft(\012)14 b Fk(C)812 1446 y Fs(2)849 1476 y Fy(.)36 b(Then)26 b(b)n(y)f(using)g(\(3-2\),)g FA([1.4])f Fy(and)h(the)h(form)n(ula)f(\()p Ft(A)2577 1446 y Fs(1)2629 1476 y Ft(\012)14 b(A)2774 1446 y Fs(2)2811 1476 y Fy(\))h Fr(o)f Fy(\()p Fu(X)3045 1446 y Fs(1)3096 1476 y Ft(\002)f Fu(X)3250 1446 y Fs(2)3287 1476 y Fy(\))3342 1454 y Ft(\030)3342 1480 y Fy(=)418 1576 y(\()p Ft(A)516 1546 y Fs(1)572 1576 y Fr(o)18 b Fu(X)731 1546 y Fs(1)768 1576 y Fy(\))h Ft(\012)f Fy(\()p Ft(A)1000 1546 y Fs(2)1056 1576 y Fr(o)g Fu(X)1215 1546 y Fs(2)1252 1576 y Fy(\))27 b(\(cf.)38 b([12]\))27 b(one)g(obtains)g(easily)g(the)h(monomorphism)418 1841 y(\(3-3\))151 b Fk(C)p Fu(=)p Fk(K)23 b Fu(,)-14 b Ft(!)1031 1699 y Fg( )1142 1762 y(Y)1097 1944 y Fq(a)1133 1927 y Fe(2)1166 1944 y Fn(2)p Fq(A)1261 1927 y Fe(2)1306 1841 y Fk(C)1357 1807 y Fs(1)1413 1841 y Ft(\012)18 b Fy([)p Fu(C)6 b Fy(\()p Fu(Z)1679 1807 y Fs(2)1673 1863 y Fq(a)1709 1847 y Fe(2)1746 1841 y Fy(\))19 b Fr(o)f Fu(X)1956 1807 y Fs(2)1992 1841 y Fy(])2015 1699 y Fg(!)2099 1841 y Ft(\002)2182 1699 y Fg( )2293 1762 y(Y)2248 1944 y Fq(a)2284 1927 y Fe(1)2317 1944 y Fn(2)p Fq(A)2412 1927 y Fe(1)2444 1841 y Fy([)p Fu(C)6 b Fy(\()p Fu(Z)2627 1807 y Fs(1)2621 1863 y Fq(a)2657 1847 y Fe(1)2694 1841 y Fy(\))18 b Fr(o)g Fu(X)2903 1807 y Fs(1)2940 1841 y Fy(])g Ft(\012)g Fk(C)3115 1807 y Fs(2)3152 1699 y Fg(!)3232 1841 y Fu(:)418 2134 y Fy(This)32 b(is)g(supp)r(osed)g(to)g(b)r(e)g(a)g(progress,)f(since)g Fu(C)6 b Fy(\()p Fu(Z)2080 2094 y Fq(j)2074 2163 y(a)2110 2146 y Fm(j)2146 2134 y Fy(\))21 b Fr(o)g Fu(X)2361 2104 y Fq(j)2428 2134 y Fy(is)32 b(usually)f(simpler)h(than)g Fk(C)3349 2104 y Fq(j)3384 2134 y Fy(.)418 2240 y(If)g(one)f(kno)n(ws)g(explicitly)h(the)g (most)f(re\014ned)h(in)n(v)-5 b(arian)n(t)30 b(decomp)r(ositions)h Ft(D)2912 2210 y Fq(j)2910 2260 y Fs(m)p Fq(ax)3072 2240 y Ft(2)g Fr(D)14 b Fy(\()p Ft(Z)3311 2210 y Fq(j)3352 2240 y Fy(\),)418 2340 y Fu(j)28 b Fy(=)23 b(1)p Fu(;)14 b Fy(2,)26 b(a)i(go)r(o)r(d)f(idea)g(is)g(to)h(use)f(them)h(in)g(the)g(relations)f(ab)r (o)n(v)n(e.)518 2439 y(In)36 b(fact,)j(\(3-3\))c(is)h(the)h(main)f(structure) g(result.)62 b(W)-7 b(e)37 b(shall)e(not)h(giv)n(e)g(the)g(form)n(ula)g(for) 418 2539 y(the)31 b(essen)n(tial)e(sp)r(ectrum)i(in)g(this)g(generalit)n(y)-7 b(.)44 b(The)30 b(next)h(paragraph)d(presen)n(ts)h(in)i(detail)f(a)418 2638 y(relev)-5 b(an)n(t)27 b(particular)f(case.)418 2786 y FA([3.2])k(A)i(simple)g(case.)518 2885 y Fy(Man)n(y)h(sp)r(ecial)h(instances) g(of)g(cartesian)f(anisotrop)n(y)f(ma)n(y)h(b)r(e)i(studied.)57 b(In)34 b(particular,)418 2993 y(the)25 b(t)n(w)n(o)f(compacti\014cations) 1400 2968 y(^)1365 2993 y Fu(X)1441 2969 y Fs(1)1502 2993 y Fy(and)1696 2968 y(^)1660 2993 y Fu(X)1736 2969 y Fs(2)1798 2993 y Fy(ma)n(y)g(ha)n(v)n(e)f("di\013eren)n(t)i(natures".)34 b(Let)25 b(us)g(restrict)418 3093 y(ourselv)n(es)31 b(here)h(to)h(a)f (simple,)j(though)d(not)h(trivial)f(case,)h(for)g(whic)n(h)f(the)i(form)n (ula)d(for)i(the)418 3193 y(essen)n(tial)27 b(sp)r(ectrum)h(is)f(quite)h (explicit.)518 3296 y(F)-7 b(or)32 b Fu(j)k Fy(=)31 b(1)p Fu(;)14 b Fy(2,)33 b(let)1164 3275 y(^)1140 3296 y Fu(X)1216 3265 y Fq(j)1282 3296 y Fy(:=)1425 3275 y(~)1401 3296 y Fu(X)1477 3265 y Fq(j)1545 3296 y Fy(b)r(e)g(the)g(sp)r(ectrum)g(of)f(the)h(C*-algebra) d Fu(V)19 b(O)r Fy(\()p Fu(X)3091 3265 y Fq(j)3126 3296 y Fy(\).)53 b(Note)418 3422 y(that)646 3397 y(~)611 3422 y Fu(X)687 3398 y Fs(1)750 3422 y Ft(\002)877 3397 y Fy(~)842 3422 y Fu(X)918 3398 y Fs(2)994 3422 y Fy(and)1235 3399 y Fr(^)1169 3422 y Fu(X)1245 3398 y Fs(1)1300 3422 y Ft(\002)18 b Fu(X)1459 3398 y Fs(2)1535 3422 y Fy(di\013ers)41 b(a)e(lot.)75 b(While)41 b(the)g(b)r(oundary)e Fu(R)i Fy(of)3147 3399 y Fr(^)3080 3422 y Fu(X)3156 3398 y Fs(1)3211 3422 y Ft(\002)18 b Fu(X)3370 3398 y Fs(2)418 3528 y Fy(consists)29 b(only)g(of)g(\014xed)h(p)r(oin)n(ts)g (for)f(the)g(translations,)g(in)h Fu(Z)i Fy(:=)26 b(\()2583 3502 y(~)2547 3528 y Fu(X)2623 3504 y Fs(1)2679 3528 y Ft(\002)2799 3502 y Fy(~)2764 3528 y Fu(X)2840 3504 y Fs(2)2876 3528 y Fy(\))20 b Ft(n)f Fu(X)3065 3498 y Fs(1)3120 3528 y Ft(\002)f Fu(X)3279 3498 y Fs(2)3342 3528 y Fy(=)418 3628 y(\()p Fu(X)526 3597 y Fs(1)580 3628 y Ft(\002)f Fu(R)726 3597 y Fs(2)763 3628 y Fy(\))g Ft(t)h Fy(\()p Fu(R)981 3597 y Fs(1)1035 3628 y Ft(\002)f Fu(X)1193 3597 y Fs(2)1230 3628 y Fy(\))g Ft(t)h Fy(\()p Fu(R)1448 3597 y Fs(1)1502 3628 y Ft(\002)f Fu(R)1648 3597 y Fs(2)1685 3628 y Fy(\))27 b(only)f(the)i(p)r(oin)n(ts)f(of)g Fu(R)2475 3597 y Fs(1)2529 3628 y Ft(\002)17 b Fu(R)2675 3597 y Fs(2)2738 3628 y Fy(are)26 b(\014xed.)37 b(Another)418 3727 y(w)n(a)n(y)d(to)h(form)n (ulate)f(this)i(is)f(to)g(sa)n(y)f(that)h Fu(V)19 b(O)r Fy(\()p Fu(X)2032 3697 y Fs(1)2070 3727 y Fy(\))k Ft(\012)g Fu(V)c(O)r Fy(\()p Fu(X)2453 3697 y Fs(2)2491 3727 y Fy(\))35 b(and)g Fu(V)19 b(O)r Fy(\()p Fu(X)2967 3697 y Fs(1)3028 3727 y Ft(\002)k Fu(X)3192 3697 y Fs(2)3228 3727 y Fy(\))36 b(are)418 3827 y(quite)28 b(di\013eren)n(t.)518 3926 y(F)-7 b(or)25 b(eac)n(h)g Fu(j)5 b Fy(,)26 b(w)n(e)g(c)n(ho)r(ose)e(the)j(greatest)d(in)n(v)-5 b(arian)n(t)25 b(decomp)r(osition)g Ft(D)2732 3896 y Fq(j)2730 3947 y Fs(max)2880 3926 y Fy(:=)e(\()p Ft(f)p Fu(r)3104 3896 y Fq(j)3139 3926 y Ft(g)p Fy(\))3213 3942 y Fq(r)3246 3926 y Fm(j)3277 3942 y Fn(2)p Fq(R)3372 3926 y Fm(j)418 4034 y Fy(of)32 b Ft(R)587 4004 y Fq(j)652 4034 y Fy(:=)770 3967 y Fg(\000)808 4034 y Fu(R)872 4004 y Fq(j)906 4034 y Fu(;)14 b(\013)996 4004 y Fq(j)1032 4034 y Fu(;)g(X)1145 4004 y Fq(j)1179 3967 y Fg(\001)1217 4034 y Fy(.)49 b(Then)32 b(the)g(C*-algebras)d Fu(C)6 b Fy(\()p Fu(Z)2274 3994 y Fq(j)2268 4063 y(a)2304 4046 y Fm(j)2340 4034 y Fy(\))32 b(in)g(\(3-3\))f(coincide)g(all)h(with)g Fr(C)418 4134 y Fy(and)c(one)f(gets)g(immediately)418 4336 y(\(3-4\))418 b Fk(C)p Fu(=)p Fk(K)23 b Fu(,)-14 b Ft(!)1342 4257 y Fg(Y)1299 4439 y Fq(r)1332 4422 y Fe(2)1364 4439 y Fn(2)p Fq(R)1459 4422 y Fe(2)1505 4336 y Fu(C)1584 4269 y Fg(\002)1619 4336 y Fy(\()p Fu(X)1727 4302 y Fs(2)1764 4336 y Fy(\))1796 4302 y Fq(])1827 4336 y Fy(;)14 b Fk(C)1915 4302 y Fs(1)1952 4269 y Fg(\003)2005 4336 y Ft(\002)2131 4257 y Fg(Y)2088 4439 y Fq(r)2121 4422 y Fe(1)2153 4439 y Fn(2)p Fq(R)2248 4422 y Fe(1)2294 4336 y Fu(C)2373 4269 y Fg(\002)2408 4336 y Fy(\()p Fu(X)2516 4302 y Fs(1)2553 4336 y Fy(\))2585 4302 y Fq(])2616 4336 y Fy(;)g Fk(C)2704 4302 y Fs(2)2741 4269 y Fg(\003)2789 4336 y Fu(:)518 4608 y Fy(T)-7 b(o)22 b(apply)h(Theorem)f(1.8.)35 b(w)n(e)22 b(need)h(some)g(preparations.)33 b(W)-7 b(e)24 b(shall)e(w)n(ork)g (in)h(the)g(Hilb)r(ert)418 4708 y(space)e Ft(H)j Fy(:=)f Fu(L)896 4678 y Fs(2)932 4708 y Fy(\()p Fu(X)7 b Fy(\),)23 b(whic)n(h)f(w)n(e)f(iden)n (tify)h(with)g Ft(H)2016 4678 y Fs(1)2060 4708 y Ft(\012)7 b(H)2203 4678 y Fs(2)2239 4708 y Fy(,)23 b Ft(H)2356 4678 y Fq(j)2414 4708 y Fy(:=)g Fu(L)2582 4678 y Fs(2)2619 4708 y Fy(\()p Fu(X)2727 4678 y Fq(j)2761 4708 y Fy(\).)36 b(Let)21 b Fu(h)i Fy(:)g Fu(X)3187 4678 y Fq(])3241 4708 y Ft(!)g Fr(R)418 4808 y Fy(b)r(e)k(a)g(con)n(tin)n(uous,)f(prop)r(er)f(function)j(and)e Fu(V)46 b Fy(a)26 b(real)g(elemen)n(t)h(of)g Fu(V)19 b(O)r Fy(\()p Fu(X)2749 4777 y Fs(1)2786 4808 y Fy(\))e Ft(\012)g Fu(V)h(O)r Fy(\()p Fu(X)3156 4777 y Fs(2)3194 4808 y Fy(\).)37 b(Let)418 4907 y(us)e(consider)e(an)i(elemen)n(t)g Fu(Z)1356 4919 y Fq(a)1430 4907 y Fy(of)g(the)g(cartesian)f(in)n(v)-5 b(arian)n(t)33 b(decomp)r(osition,)j(for)e(example)418 5013 y Ft(f)p Fu(r)499 4983 y Fs(1)537 5013 y Ft(g)13 b(\002)705 4987 y Fy(~)670 5013 y Fu(X)746 4989 y Fs(2)807 5013 y Fy(for)25 b(some)f Fu(r)1176 4983 y Fs(1)1237 5013 y Ft(2)g Fu(R)1380 4983 y Fs(1)1416 5013 y Fy(.)37 b(W)-7 b(e)25 b(iden)n(tify)h(in)f(an)g(ob)n (vious)e(w)n(a)n(y)h Fu(C)6 b Fy(\()p Ft(f)p Fu(r)2764 4983 y Fs(1)2802 5013 y Ft(g)13 b(\002)2971 4987 y Fy(~)2935 5013 y Fu(X)3011 4989 y Fs(2)3048 5013 y Fy(\))25 b(with)h(the)418 5112 y(C*-subalgebra)20 b Ft(C)1007 5082 y Fs(2)1067 5112 y Fy(of)j Fu(B)t(C)1283 5124 y Fq(u)1326 5112 y Fy(\()p Fu(X)7 b Fy(\))23 b(consisting)f(of)h(functions)g(of)g(the)g(form)f Fu(')p Fy(\()p Fu(x)2863 5082 y Fs(1)2902 5112 y Fu(;)14 b(x)2986 5082 y Fs(2)3023 5112 y Fy(\))23 b(=)g Fu(')3220 5082 y Fs(2)3258 5112 y Fy(\()p Fu(x)3337 5082 y Fs(2)3375 5112 y Fy(\))p eop %%Page: 17 17 bop 1026 314 a Fw(C*{ALGEBRAS)29 b(AND)f(THE)g(ESSENTIAL)g(SPECTR)n(UM)565 b(17)418 513 y Fy(with)31 b Fu(')664 483 y Fs(2)730 513 y Ft(2)d Fu(V)19 b(O)r Fy(\()p Fu(X)1053 483 y Fs(2)1090 513 y Fy(\).)47 b(Then)30 b(the)h(function)h Fu(V)1952 483 y Fs(2)1934 539 y Fq(r)1967 523 y Fe(1)2003 513 y Fy(\()p Fu(x)2082 483 y Fs(2)2120 513 y Fy(\))c(:=)2309 492 y(^)2296 513 y Fu(V)19 b Fy(\()p Fu(r)2434 483 y Fs(1)2472 513 y Fu(;)14 b(x)2556 483 y Fs(2)2594 513 y Fy(\))31 b(\()2702 492 y(^)2689 513 y Fu(V)50 b Fy(is)30 b(the)h(con)n(tin)n(uous)418 629 y(extension)k(of)g Fu(V)54 b Fy(to)1130 608 y(^)1106 629 y Fu(X)43 b Fy(=)1353 603 y(~)1318 629 y Fu(X)1394 605 y Fs(1)1454 629 y Ft(\002)1578 603 y Fy(~)1542 629 y Fu(X)1618 605 y Fs(2)1655 629 y Fy(\))35 b(is)g(an)g(elemen)n(t)h(of)f Ft(C)2402 599 y Fs(2)2439 629 y Fy(.)60 b(In)35 b(a)g(similar)f(w)n(a)n(y)g (w)n(e)h(get)418 729 y(elemen)n(ts)30 b Fu(V)827 698 y Fs(1)808 755 y Fq(r)841 738 y Fe(2)905 729 y Ft(2)e(C)1037 698 y Fs(1)1102 729 y Fy(:=)i(the)g(C*-subalgebra)e(of)i Fu(B)t(C)2136 741 y Fq(u)2180 729 y Fy(\()p Fu(X)7 b Fy(\))30 b(consisting)g(of)g(functions)g (of)h(the)418 831 y(form)c Fu(')p Fy(\()p Fu(x)747 801 y Fs(1)786 831 y Fu(;)14 b(x)870 801 y Fs(2)907 831 y Fy(\))24 b(=)e Fu(')1104 801 y Fs(1)1142 831 y Fy(\()p Fu(x)1221 801 y Fs(1)1259 831 y Fy(\))28 b(with)g Fu(')1562 801 y Fs(1)1622 831 y Ft(2)c Fu(V)19 b(O)r Fy(\()p Fu(X)1941 801 y Fs(1)1978 831 y Fy(\).)518 931 y(Then)28 b(b)n(y)f(a)g(straigh)n(tforw)n(ard)e(application)i(of)g (Theorem)g(1.8.)36 b(w)n(e)27 b(get)418 1075 y FA(Corollary)38 b(3.2.)47 b Fj(The)31 b(self-adjoin)n(t)g(op)r(erator)f Fu(H)36 b Fy(:=)29 b Fu(h)p Fy(\()p Fu(P)12 b Fy(\))22 b(+)e Fu(V)51 b Fj(is)31 b(a\016liated)g(to)h(the)g(C*-)418 1175 y(subalgebra)837 1107 y Fg(\012)876 1175 y Fy([)p Fu(V)19 b(O)r Fy(\()p Fu(X)1139 1144 y Fs(1)1176 1175 y Fy(\))g Ft(\012)f Fu(V)h(O)r Fy(\()p Fu(X)1550 1144 y Fs(2)1587 1175 y Fy(\)])g Ft(\001)g Fu(C)1762 1187 y Fs(0)1799 1175 y Fy(\()p Fu(X)1907 1144 y Fq(])1938 1175 y Fy(\))1970 1107 y Fg(\013)2040 1175 y Fj(of)30 b Fu(B)t Fy(\()p Ft(H)q Fy(\))h Fj(and)f(its)g(essen)n(tial)g(sp)r(ectrum)g(is)418 1274 y(giv)n(en)d(b)n(y)418 1563 y Fy(\(3-5\))225 b Fu(\033)-52 b(\033)812 1559 y(\033)863 1575 y Fs(ess)949 1563 y Fy(\()p Fu(H)7 b Fy(\))23 b(=)p 1200 1383 845 4 v 1200 1396 a Fg(0)1200 1545 y(@)1323 1484 y([)1273 1669 y Fq(r)1306 1652 y Fe(1)1338 1669 y Fn(2)p Fq(R)1433 1649 y Fe(1)1433 1687 y(0)1479 1563 y Fu(\033)s Fy([)p Fu(h)p Fy(\()p Fu(P)12 b Fy(\))19 b(+)f Fu(V)1898 1534 y Fs(2)1879 1591 y Fq(r)1912 1574 y Fe(1)1949 1563 y Fy(])1972 1396 y Fg(1)1972 1545 y(A)2063 1563 y Ft([)p 2137 1383 V 2137 1396 a Fg(0)2137 1545 y(@)2260 1484 y([)2209 1669 y Fq(r)2242 1652 y Fe(2)2274 1669 y Fn(2)p Fq(R)2369 1649 y Fe(2)2369 1687 y(0)2416 1563 y Fu(\033)s Fy([)p Fu(h)p Fy(\()p Fu(P)12 b Fy(\))19 b(+)f Fu(V)2835 1534 y Fs(1)2816 1591 y Fq(r)2849 1574 y Fe(2)2886 1563 y Fy(])2909 1396 y Fg(1)2909 1545 y(A)2981 1563 y Fu(;)418 1859 y Fj(where)27 b Fu(R)722 1819 y Fq(j)721 1881 y Fs(0)786 1859 y Fj(is)h(some)f(arbitrary)e(dense)j (subset)f(of)h Fu(R)2071 1829 y Fq(j)2105 1859 y Fj(.)518 2004 y Fy(This)f(result)h(is)f(quite)h(di\013eren)n(t)g(in)g(nature)f(from)g (\(2-1\).)518 2103 y(One)j(can)g(giv)n(e)g(a)g(more)g(detailed)h(v)n(ersion)e (of)i(\(3-5\),)f(b)n(y)h(taking)f(pro\014t)g(of)h(the)g(fact)g(that)418 2203 y(the)38 b(functions)g Fu(V)1006 2173 y Fs(2)987 2229 y Fq(r)1020 2213 y Fe(1)1094 2203 y Fy(and)f Fu(V)1332 2173 y Fs(1)1313 2229 y Fq(r)1346 2213 y Fe(2)1421 2203 y Fy(do)g(not)g(dep)r(end) i(of)e(all)g(the)h(v)-5 b(ariables.)65 b(W)-7 b(e)38 b(shall)f(do)h(this)418 2302 y(only)d(for)f(an)g(in)n(teresting)g(particular)g(case.)57 b(Let)35 b(us)g(consider)f(Sc)n(hr\177)-42 b(odinger)33 b(op)r(erators)g(in) 418 2402 y Fu(X)k Fy(:=)30 b Fr(R)696 2372 y Fq(n)737 2380 y Fe(1)776 2372 y Fs(+)p Fq(n)868 2380 y Fe(2)935 2402 y Fy(=)g Fu(X)1106 2372 y Fs(1)1164 2402 y Ft(\002)21 b Fu(X)1326 2372 y Fs(2)1363 2402 y Fy(,)33 b(i.e.)51 b Fu(h)p Fy(\()p Fu(P)12 b Fy(\))31 b(=)f(\001)h(=)f Fu(P)2139 2372 y Fs(2)2207 2402 y Fy(=)g Ft(\000r)2436 2372 y Fs(2)2504 2402 y Fy(=)g(\001)2668 2372 y Fs(1)2727 2402 y Ft(\012)21 b Fy(1)g(+)g(1)g Ft(\012)g Fy(\001)3180 2372 y Fs(2)3218 2402 y Fy(.)51 b(W)-7 b(e)418 2502 y(in)n(terpret)27 b(no)n(w)f Fu(V)1001 2472 y Fq(k)982 2529 y(r)1015 2513 y Fm(j)1077 2502 y Fy(as)g(a)h(function)g(in)h Fu(X)1744 2472 y Fq(k)1807 2502 y Fy(=)22 b Fr(R)1948 2472 y Fq(n)1989 2481 y Fm(k)2063 2502 y Fy(and)27 b(set)g Fu(H)2429 2472 y Fq(k)2422 2529 y(r)2455 2513 y Fm(j)2512 2502 y Fy(:=)c(\001)2692 2472 y Fq(k)2750 2502 y Fy(+)17 b Fu(V)2899 2472 y Fq(k)2880 2529 y(r)2913 2513 y Fm(j)2975 2502 y Fy(\(self-adjoin)n(t)418 2608 y(op)r(erator)26 b(in)i Ft(H)921 2578 y Fq(k)962 2608 y Fy(\))g(and)f Fu(b)1219 2578 y Fq(k)1219 2635 y(r)1252 2619 y Fm(j)1309 2608 y Fy(:=)c(inf)q(\()p Fu(H)1623 2578 y Fq(k)1616 2635 y(r)1649 2619 y Fm(j)1684 2608 y Fy(\),)28 b Fu(j;)14 b(k)26 b Fy(=)d(1)p Fu(;)14 b Fy(2,)26 b Fu(j)i Ft(6)p Fy(=)23 b Fu(k)s Fy(.)37 b(Then)27 b(w)n(e)h(ha)n(v)n(e)1225 2841 y Fu(\033)-51 b(\033)1219 2837 y(\033)1269 2853 y Fs(ess)1355 2841 y Fy(\()p Fu(H)7 b Fy(\))24 b(=)1606 2724 y Fg(\024)1650 2841 y Fy(min)1802 2724 y Fg(\032)1865 2841 y Fy(inf)1883 2898 y Fq(r)1916 2882 y Fe(1)1980 2841 y Fu(b)2016 2807 y Fs(2)2016 2864 y Fq(r)2049 2847 y Fe(1)2084 2841 y Fu(;)14 b Fy(inf)2139 2898 y Fq(r)2172 2882 y Fe(2)2236 2841 y Fu(b)2272 2807 y Fs(1)2272 2864 y Fq(r)2305 2847 y Fe(2)2341 2724 y Fg(\033)2417 2841 y Fu(;)g Ft(1)2537 2724 y Fg(\023)418 3068 y Fy(as)33 b(a)g(consequence)g(of) g(\(3-5\))g(and)h(w)n(ell-kno)n(wn)e(facts)h(ab)r(out)h(the)g(sp)r(ectrum)f (of)h(factorized)418 3168 y(op)r(erators.)518 3267 y(By)e(ob)n(vious)e (particularizations)h(one)g(gets)h(Example)f(0.2.)50 b(of)32 b(the)h(In)n(tro)r(duction.)50 b(One)418 3367 y(has)30 b(to)g(tak)n(e)g(in)n (to)g(accoun)n(t)f(the)i(sp)r(eci\014cit)n(y)g(of)f(the)h(case)e Fu(n)2342 3379 y Fs(1)2407 3367 y Fy(=)e Fu(n)2549 3379 y Fs(2)2614 3367 y Fy(=)g(1)j(and)g(the)h(fact)g(that)418 3466 y(functions)d(with)g (radial)f(limits)h(ha)n(v)n(e)e(the)i(v)-5 b(anishing)27 b(oscillation)g (prop)r(ert)n(y)-7 b(.)1205 3656 y Fz(Appendix)31 b(A.)e(Graded)i (C*-algebras)518 3805 y Fy(The)f(imp)r(ortance)g(of)g(graded)f(C*-algebras)e (in)j(the)h(generalized)e Fu(N)9 b Fy(-b)r(o)r(dy)29 b(problem)h(w)n(as)418 3905 y(put)d(in)n(to)f(evidence)h(in)f([1])g(and)h([3].)36 b(W)-7 b(e)27 b(do)f(not)g(in)n(tend)h(to)f(discuss)g(this)h(concept)f(in)h (detail.)418 4004 y(Our)g(purp)r(ose)f(will)i(b)r(e)g(just)g(to)f(sho)n(w)f (ho)n(w)h(the)h(basic)f(tec)n(hnical)g(result)g(\(A-2\))g(follo)n(ws)f(from) 418 4104 y(Prop)r(osition)i(1.5.\(a\).)41 b(W)-7 b(e)30 b(tak)n(e)e(pro\014t) h(of)h(the)f(o)r(ccasion)f(to)h(in)n(tro)r(duce)g(some)g(new)g(ob)5 b(jects)418 4204 y(in)28 b(graded)e(C*-algebras.)518 4303 y(Let)i(\()p Ft(L)p Fu(;)14 b Ft(\024)p Fy(\))29 b(a)g(\014nite)g(lattice,)g(with)g Fk(0)24 b Fy(:=)h(min)p Ft(L)g(6)p Fy(=)f Fk(1)h Fy(:=)f(max)p Ft(L)p Fy(.)40 b(F)-7 b(or)28 b Fu(b)c Ft(2)h(L)p Fy(,)30 b Ft(M)24 b(\032)g(L)29 b Fy(w)n(e)418 4403 y(sa)n(y)23 b(that)i Fu(b)g Fv(is)i(under)d Ft(M)h Fy(and)f(w)n(e)g(write)h Fu(b)d Ff(4)h Ft(M)h Fy(if)h(there)g(is)f(some)g Fu(a)f Ft(2)g(M)i Fy(suc)n(h)f(that)h Fu(b)d Ft(\024)h Fu(a)p Fy(.)418 4503 y(Ob)n(viously)29 b Fk(0)e Ff(4)f Ft(M)k Fy(for)g(an)n(y)f Ft(M)p Fy(.)44 b(If)30 b Fu(b)1678 4515 y Fs(1)1742 4503 y Ft(\024)d Fu(b)1870 4515 y Fs(2)1934 4503 y Ff(4)g Ft(M)i Fy(then)i Fu(b)2383 4515 y Fs(1)2447 4503 y Ff(4)c Ft(M)j Fy(and)f(if)i Fu(b)c Ff(4)f Ft(M)h(\032)g(M)3384 4472 y Fn(0)418 4602 y Fy(then)h Fu(b)23 b Ff(4)g Ft(M)854 4572 y Fn(0)877 4602 y Fy(.)518 4702 y(W)-7 b(e)27 b(sa)n(y)f(that)h Ft(M)22 b(2)i Fy(2)1225 4672 y Fn(L)1301 4702 y Fv(is)30 b(down-stable)d Fy(and)g(w)n(e)f(write)h Ft(M)c(2)g Fu(D)r(S)5 b Fy(\()p Ft(L)p Fy(\))28 b(if)f Fu(b)c Ff(4)f Ft(M)27 b Fy(implies)418 4801 y Fu(b)c Ft(2)g(M)p Fy(.)37 b(This)28 b(is)f(equiv)-5 b(alen)n(t)28 b(to)f Ft(M)c Fy(=)g Ft(M)1795 4771 y Fs(do)n(wn)1951 4801 y Fy(,)28 b(with)g Ft(M)2291 4771 y Fs(do)n(wn)2470 4801 y Fy(:=)23 b Ft(f)p Fu(b)f Ft(2)h(L)h(j)f Fu(b)g Ff(4)g Ft(Mg)p Fy(.)36 b(Note)418 4901 y(that)f(\()p Ft([)692 4913 y Fq(j)728 4901 y Ft(M)828 4913 y Fq(j)863 4901 y Fy(\))895 4871 y Fs(do)n(wn)1086 4901 y Fy(=)g Ft([)1241 4913 y Fq(j)1276 4901 y Ft(M)1376 4871 y Fs(do)n(wn)1376 4923 y Fq(j)1567 4901 y Fy(and)g(\()p Ft(\\)1823 4913 y Fq(j)1858 4901 y Ft(M)1958 4913 y Fq(j)1993 4901 y Fy(\))2025 4871 y Fs(do)n(wn)2217 4901 y Ft(\032)f(\\)2371 4913 y Fq(j)2407 4901 y Ft(M)2507 4871 y Fs(do)n(wn)2507 4923 y Fq(j)2697 4901 y Fy(\(strictly)h(in)g(general:)418 5013 y(tak)n(e)29 b Ft(M)700 5025 y Fs(1)757 5013 y Ft(\\)20 b(M)932 5025 y Fs(2)995 5013 y Fy(=)26 b Ft(;)p Fy(\).)43 b(The)30 b(sets)f Ft(f)p Fu(a)p Ft(g)1691 4983 y Fs(do)n(wn)1876 5013 y Fy(are)g(sublattices)g(and)h(w)n(ere) e(denoted)i(in)g([1])f(b)n(y)418 5112 y Ft(L)475 5124 y Fq(a)516 5112 y Fy(.)36 b(W)-7 b(e)28 b(ha)n(v)n(e:)p eop %%Page: 18 18 bop 418 314 a Fw(18)1076 b(MARIUS)28 b(M)1953 297 y(\025)1944 314 y(ANTOIU)418 513 y FA(Lemma)33 b(A.1.)47 b Fu(D)r(S)5 b Fy(\()p Ft(L)p Fy(\))29 b Fj(is)g(a)g(Bo)r(ole)f(subalgebra)f(of)i Fy(\(2)2232 483 y Fn(L)2282 513 y Fu(;)14 b Ft(\032)p Fy(\))28 b Fj(whic)n(h)h(includes)g(canonically)418 613 y Ft(L)p Fj(.)36 b(The)23 b(application)g Fu(d)g Fy(:)h(2)1280 582 y Fn(L)1352 613 y Ft(!)f Fy(2)1500 582 y Fn(L)1550 613 y Fj(,)h Fu(d)p Fy(\()p Ft(M)p Fy(\))g(:=)f Ft(M)2039 582 y Fs(do)n(wn)2218 613 y Fj(is)g(an)g(increasing,)g(idemp)r(oten)n(t)h(map)418 712 y(with)k(range)e Fu(D)r(S)5 b Fy(\()p Ft(L)p Fy(\))p Fj(.)518 864 y Fy(Let)29 b(no)n(w)f Fk(C)c Fy(=)1007 802 y Fg(P)1094 889 y Fq(a)p Fn(2L)1239 864 y Fk(C)p Fy(\()p Fu(a)p Fy(\))29 b(b)r(e)g(a)f Ft(L)p Fy(-)p Fv(gr)l(ade)l(d)k(C*-algebr)l(a)p Fy(.)42 b(This)29 b(means)f(that)h(the)g(sum)g(is)418 972 y(direct)24 b(and)h(that)f Fk(C)p Fy(\()p Fu(a)p Fy(\))p Fk(C)p Fy(\()p Fu(b)p Fy(\))g Ft(\032)f Fk(C)p Fy(\()p Fu(a)12 b Ft(_)g Fu(b)p Fy(\).)36 b(F)-7 b(or)24 b(an)n(y)g Ft(M)e(2)i Fy(2)2283 941 y Fn(L)2357 972 y Fy(let)h(us)f(set)h Fk(K)2759 984 y Fn(M)2864 972 y Fy(:=)2975 909 y Fg(P)3063 997 y Fq(b)7 b(=)-41 b Fn(2M)3233 972 y Fk(C)p Fy(\()p Fu(b)p Fy(\).)418 1071 y(This)31 b(is)g(a)f(closed)g (subspace)h(of)f Fk(C)h Fy(ha)n(ving)f Fk(C)1870 1083 y Fn(M)1981 1071 y Fy(:=)2097 1009 y Fg(P)2184 1096 y Fq(b)p Fn(2M)2354 1071 y Fk(C)p Fy(\()p Fu(b)p Fy(\))h(as)g(a)f(supplemen)n(t,)i(but)g(in)418 1171 y(order)g(that)i Fk(K)882 1183 y Fn(M)999 1171 y Fy(b)r(e)g(an)f(ideal)h (w)n(e)f(ha)n(v)n(e)g(to)g(imp)r(ose)h(the)g(restriction)f Ft(M)g(2)g Fu(D)r(S)5 b Fy(\()p Ft(L)p Fy(\).)56 b(The)418 1271 y(critical)27 b(relation,)g(v)n(ery)f(easy)h(to)h(obtain,)f(is)h Ft(\\)1910 1283 y Fq(j)1945 1271 y Fk(K)2000 1283 y Fn(M)2078 1291 y Fm(j)2137 1271 y Fy(=)23 b Fk(K)2280 1283 y Fn([)2325 1291 y Fm(j)2356 1283 y Fn(M)2434 1291 y Fm(j)2497 1271 y Fy(if)28 b(ev)n(ery)e Ft(M)2892 1283 y Fq(j)2955 1271 y Fy(is)h(in)h Fu(D)r(S)5 b Fy(\()p Ft(L)p Fy(\).)418 1371 y(By)27 b(applying)g(Prop)r (osition)f(1.5.\(a\))h(w)n(e)g(get)418 1523 y FA(Corollary)34 b(A.2.)47 b Fj(Set)28 b Ft(L)1264 1492 y Fn(0)1310 1523 y Fy(:=)23 b Ft(L)c(n)f(f)p Fk(1)p Ft(g)26 b Fj(and)h Fk(K)d Fy(:=)f Fk(K)2115 1535 y Fn(L)2161 1518 y Fh(0)2211 1523 y Fy(=)f Fk(C)p Fy(\()p Fk(1)p Fy(\))28 b Fj(the)g(minimal)g(ideal.)37 b(Then)418 1741 y Fy(\(A-1\))959 b Fk(C)p Fu(=)p Fk(K)23 b Fu(,)-14 b Ft(!)1865 1662 y Fg(Y)1859 1840 y Fq(j)s Fn(2)p Fq(J)1991 1741 y Fk(C)p Fu(=)p Fk(K)2139 1753 y Fn(M)2217 1761 y Fm(j)418 2033 y Fj(for)27 b(an)n(y)f(family)h Fy(\()p Ft(M)1086 2045 y Fq(j)1121 2033 y Fy(\))1153 2045 y Fq(j)s Fn(2)p Fq(J)1298 2033 y Ft(\032)c Fu(D)r(S)5 b Fy(\()p Ft(L)p Fy(\))28 b Fj(suc)n(h)f(that)g Ft([)2083 2045 y Fq(j)s Fn(2)p Fq(J)2205 2033 y Ft(M)2305 2045 y Fq(j)2363 2033 y Fy(=)c Ft(L)2508 2002 y Fn(0)2558 2033 y Fj(and)k Ft(M)2819 2045 y Fq(j)2881 2033 y Fj(is)g(not)g(a)g(subset)418 2132 y(of)h Ft(M)613 2144 y Fq(j)643 2128 y Fh(0)697 2132 y Fj(if)h Fu(j)f Ft(6)p Fy(=)22 b Fu(j)962 2102 y Fn(0)985 2132 y Fj(.)518 2284 y Fy(Of)27 b(course,)g(in)h(applications)e Fk(K)i Fy(will)g(b)r(e)g(\(isomorphic)f(to\))h(the)f(ideal)h(of)f(compact)g (op)r(era-)418 2384 y(tors.)41 b(There)28 b(is)h(still)g(a)g(reason)f(of)h (discon)n(ten)n(temen)n(t)f(in)i(the)f(w)n(a)n(y)f(the)h(Corollary)e(is)i (form)n(u-)418 2483 y(lated,)23 b(since)g(the)f(quotien)n(ts)g(in)h(the)f (righ)n(t-hand)f(side)h(are)g(not)g(iden)n(ti\014ed)h(to)f(C*-subalgebras)418 2583 y(of)k Fk(C)p Fy(.)36 b(Let)26 b(us)f(observ)n(e)f(that)i Fk(C)1395 2595 y Fn(M)1503 2583 y Fy(is)f(a)h(C*-subalgebra)d(if)j Ft(M)f Fy(is)h Ft(_)p Fy(-stable)f(and)g(in)h(particular)418 2683 y(this)34 b(is)f(v)n(eri\014ed)g(for)g Ft(M)g Fy(=)g Ft(f)p Fu(a)p Ft(g)1464 2653 y Fs(do)n(wn)1652 2683 y Fy(=)f Fu(d)p Fy(\()p Ft(f)p Fu(a)p Ft(g)p Fy(\).)55 b(W)-7 b(e)34 b(note)f(also)g(that,)i (setting)f Fu(A)g Fy(for)f(the)418 2782 y(set)27 b(of)g(all)g(coatoms)e(of)i Ft(L)g Fy(\(=)g(maximal)g(elemen)n(ts)f(in)i Ft(L)2211 2752 y Fn(0)2234 2782 y Fy(\),)2317 2715 y Fg(\000)2355 2782 y Fk(K)2410 2799 y Fn(f)p Fq(a)p Fn(g)2514 2783 y Fe(do)o(wn)2651 2715 y Fg(\001)2690 2818 y Fq(a)p Fn(2)p Fq(A)2851 2782 y Fy(is)f(a)g Fk(K)p Fy(-su\016cien)n(t)418 2893 y(family)h(of)f(ideals.)37 b(Therefore)26 b(w)n(e)h(can)h(write)f(instead)g(of)h(\(A-1\))418 3111 y(\(A-2\))752 b Fk(C)p Fu(=)p Fk(K)23 b Fu(,)-14 b Ft(!)1665 3032 y Fg(Y)1653 3210 y Fq(a)p Fn(2)p Fq(A)1797 3111 y Fk(C)1848 3128 y Fn(f)p Fq(a)p Fn(g)1952 3111 y Fe(do)o(wn)31 b Ft(\021)2212 3032 y Fg(Y)2200 3210 y Fq(a)p Fn(2)p Fq(A)2344 3111 y Fk(C)2395 3123 y Fq(a)2435 3111 y Fu(:)518 3392 y Fy(This)d(form)n(ula)f(is)h(basic)g (for)f(the)i(sp)r(ectral)e(analysis)g(of)h(self-adjoin)n(t)g(op)r(erators)e (a\016liated)418 3492 y(to)34 b(graded)f(C*-algebras,)f(the)i(N-b)r(o)r(dy)g (Hamiltonians)g(b)r(eing)g(the)g(b)r(est-kno)n(wn)f(example.)418 3591 y(It)e(giv)n(es)e(immediately)i(the)g(HWZ)g(form)n(ula)e(for)h(the)h (essen)n(tial)f(sp)r(ectrum)h(and)f(it)h(is)f(at)h(the)418 3691 y(base)c(of)h(a)f(pro)r(of)g(of)g(the)h(Mourre)f(estimate.)1670 3894 y Fz(References)461 4027 y Fw(1.)d(W.)h(Amrein,)f(A.)i(Boutet)h(de)f (Mon)n(v)n(el)g(and)g(V.)f(Georgescu,)i Fc(C)2259 4036 y Fb(0)2294 4027 y Fp(-Gr)l(oups,)i(Commutator)g(Metho)l(ds)g(and)540 4110 y(Sp)l(e)l(ctr)l(al)e(The)l(ory)f(of)g(N-Bo)l(dy)g(Hamiltonians)p Fw(,)e(Birkh\177)-35 b(auser-V)-6 b(erlag,)23 b(Basel-Boston-Berlin,)f(1996.) 461 4194 y(2.)i(W.)29 b(Amrein,)h(M.)f(M\025)-35 b(an)n(toiu)30 b(and)h(R.)e(Purice,)g Fp(Pr)l(op)l(agation)34 b(Pr)l(op)l(erties)f(for)e (Schr\177)-36 b(odinger)32 b(Op)l(er)l(ators)540 4277 y(a\016liate)l(d)26 b(to)g(Certain)f(C*-algebr)l(as)p Fw(,)f(in)g(preparation.)461 4361 y(3.)g(M.)17 b(Damak)h(and)i(V.)e(Georgescu,)h Fp(C*-algebr)l(as)j(R)l (elate)l(d)g(to)f(the)g(N-Bo)l(dy)g(Pr)l(oblem)h(and)g(the)f(Self-adjoint)540 4444 y(Op)l(er)l(ators)27 b(A\016liate)l(d)f(to)f(Them)p Fw(,)f(preprin)n(t)g (99-482)g(at)g Fc(http)c Fw(:)f Fc(==w)r(w)r(w)r(:ma:utexas:edu=mp)p Fw(-)p Fc(ar)r(c=)p Fw(.)461 4528 y(4.)24 b(J.)f(Dixmier,)e Fp(L)l(es)26 b(C*-alg)n(\022)-35 b(ebr)l(es)26 b(et)f(leurs)h(r)l(epr)l (esentations)p Fw(,)f(Gauthier-Villars,)d(P)n(aris,)g(1964.)461 4612 y(5.)i(R.)f(Ellis,)e Fp(L)l(o)l(c)l(al)t(ly)27 b(Comp)l(act)g(T)-5 b(r)l(ansformation)27 b(Gr)l(oups)p Fw(,)e(Duk)n(e)g(Math.)e(J.)h Fa(24)e Fw(\(1957\),)k(119-125.)461 4696 y(6.)e(V.)h(Georgescu)i(and)f(A.)f (Iftimo)n(vici,)f Fp(C*-A)n(lgebr)l(as)k(of)g(Ener)l(gy)g(Observables:)36 b(I.)27 b(Gener)l(al)i(The)l(ory)f(and)540 4779 y(Bumps)e(A)n(lgebr)l(as)p Fw(,)e(preprin)n(t)g(00-520)g(at)h Fc(http)19 b Fw(:)h Fc(==w)r(w)r(w)r (:ma:utexas:edu=mp)p Fw(-)p Fc(ar)r(c=)p Fw(.)461 4862 y(7.)k(E.)h(Hewitt)i (and)g(K.)e(A.)h(Ross,)f Fp(A)n(bstr)l(act)j(Harmonic)h(A)n(nalysis.)f(V)-5 b(ol.)28 b(I)p Fw(,)e(Grundlehren)h(der)f(Mathema-)540 4945 y(tisc)n(hen)e(Wissensc)n(haften,)h(v)n(ol.115,)f(Springer-V)-6 b(erlag)23 b(Berlin,)f(1979.)461 5029 y(8.)i(M.)18 b(M\025)-35 b(an)n(toiu,)19 b Fp(On)h(a)i(Class)g(of)f(A)n(nisotr)l(opic)h(Schr\177)-36 b(odinger)22 b(Op)l(er)l(ators)p Fw(,)e(to)f(app)r(ear)h(in)e(Rev.)h(Math.)g (Ph)n(ys.)540 5112 y(and)24 b(preprin)n(t)f(01-201)i(at)f Fc(http)c Fw(:)f Fc(==w)r(w)r(w)r(:ma:utexas:edu=mp)p Fw(-)p Fc(ar)r(c=)p Fw(.)p eop %%Page: 19 19 bop 1026 314 a Fw(C*{ALGEBRAS)29 b(AND)f(THE)g(ESSENTIAL)g(SPECTR)n(UM)565 b(19)461 513 y(9.)24 b(G.)33 b(K.)f(P)n(edersen,)h Fp(C*-A)n(lgebr)l(as)i (and)h(Their)e(A)n(utomorphism)j(Gr)l(oups)p Fw(,)d(Academic)f(Press,)f (London,)540 596 y(1979.)426 679 y(10.)24 b(S.)30 b(C.)h(P)n(o)n(w)n(er,)g Fp(F)-5 b(r)l(e)l(dholm)35 b(T)-5 b(o)l(eplitz)33 b(Op)l(er)l(ators)g(and)h (Slow)f(Oscil)t(lation)p Fw(,)e(Can.)g(J.)g(Math.)g Fa(32)f Fw(\(1980\),)540 762 y(1058-1071.)426 845 y(11.)24 b(M.)g(Reed)j(and)f(B.)f (Simon,)g Fp(Metho)l(ds)j(of)g(Mo)l(dern)g(Mathematic)l(al)h(Physics,)f(vol)g (IV)p Fw(,)d(Academic)h(Press,)540 928 y(London,)e(1978.)426 1011 y(12.)g(H.)18 b(T)-6 b(ak)l(ai,)20 b Fp(On)h(a)h(Duality)f(for)h(Cr)l (osse)l(d)h(Pr)l(o)l(ducts)g(of)f(C*-A)n(lgebr)l(as)p Fw(,)d(J.)g(F)-6 b(unct.)21 b(Anal.)d Fa(19)h Fw(\(1975\),)i(25-39.)418 1194 y Fx(Present)h(address:)33 b(D)1108 1189 y(\023)1108 1194 y(ep)l(ar)l(tement) 22 b(de)f(Physique)h(Th)2066 1189 y(\023)2066 1194 y(eorique,)h(Universit) 2714 1189 y(\023)2714 1194 y(e)g(de)e(Gen)3017 1189 y(\022)3017 1194 y(eve;)j(32,)d(bd.)418 1277 y(d'Yv)o(o)o(y;)k(CH-1211,)f(Gen)1193 1272 y(\022)1193 1277 y(eve)j(4,)f(Switzerland.)35 b(e-mail:)g (marius.mantoiu)p Fw(@)p Fx(physics.unige.ch)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0108070316380--