Content-Type: multipart/mixed; boundary="-------------9912210646991" This is a multi-part message in MIME format. ---------------9912210646991 Content-Type: text/plain; name="99-481.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-481.keywords" N-body problem, C*-algebras, crossed products, essential spectrum, Mourre estimate ---------------9912210646991 Content-Type: application/postscript; name="cpnb.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="cpnb.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: cpnb.dvi %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o cpnb.ps cpnb.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.11.25:1715 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false 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Fz(\025)g Fq(j)o(!1)2494 2667 y Fr(\034)2534 2682 y Fz(\025!)2625 2667 y Fs([)p Fr(H)7 b Fs(])40 b(:=)g Fr(H)3010 2681 y Fz(!)3101 2667 y Fo(exists)h(in)448 2780 y(str)-5 b(ong-r)g(esolvent)35 b(sense)e(and)1439 2992 y Fr(\033)1491 3006 y Fn(ess)1582 2992 y Fs(\()p Fr(H)7 b Fs(\))26 b(=)1955 2906 y Fl([)1857 3107 y Fz(!)r Fq(2)p Fz(X)5 b Fq(nf)p Fn(0)p Fq(g)2170 2992 y Fr(\033)s Fs(\()p Fr(H)2336 3006 y Fz(!)2386 2992 y Fs(\))p Fr(:)-1998 b Fs(\(1.1\))589 3346 y(Here)37 b Fr(\034)850 3360 y Fz(x)894 3346 y Fs([)p Fr(H)7 b Fs(])35 b(=)f(e)1208 3313 y Fz(i)p Fq(h)p Fz(x;P)10 b Fq(i)1404 3346 y Fr(H)d Fs(e)1528 3313 y Fq(\000)p Fz(i)p Fq(h)p Fz(x;P)j Fq(i)1816 3346 y Fs(and)35 b(\(e)2074 3313 y Fz(i)p Fq(h)p Fz(x;P)10 b Fq(i)2271 3346 y Fr(f)g Fs(\)\()p Fr(y)s Fs(\))35 b(=)f Fr(f)10 b Fs(\()p Fr(x)24 b Fs(+)f Fr(y)s Fs(\).)58 b(Note)37 b(that)448 3458 y Fr(H)524 3472 y Fz(!)607 3458 y Fs(dep)s(ends)30 b(only)i(on)g(the)h (one)f(dimensional)e(space)j(generated)g(b)m(y)f Fr(!)s Fs(.)47 b(The)32 b(pro)s(of)448 3571 y(of)27 b(the)f(fact)i(that)f Fr(\033)1121 3585 y Fn(ess)1212 3571 y Fs(\()p Fr(H)7 b Fs(\))27 b(is)e(equal)h(to)h(the)g(closure)f(of)g(the)h(righ)m (t-hand)e(side)g(in)g(\(1.1\))448 3684 y(is)37 b(quite)f(easy)i(\(see)g (section)f(8.4.3)i(in)d([ABG])j(and)d(note)i(that)g(the)f(sp)s(ectrum)f (of)i(a)448 3797 y(direct)h(sum,)h(not)f(necessarily)f(\014nite,)i(of)f (self-adjoin)m(t)g(op)s(erators)g(is)f(equal)g(to)i(the)448 3910 y(closure)d(of)g(the)g(union)e(of)j(the)f(sp)s(ectra\).)61 b(The)37 b(fact)h(that)f(the)g(righ)m(t-hand)f(side)g(in)448 4023 y(\(1.1\))c(is)e(closed)g(is)g(more)g(di\016cult)e(and)i(the)h (pro)s(of)e(can)i(b)s(e)f(found)f(in)g([DG].)589 4186 y(In)36 b(order)h(to)g(get)h(a)f(v)m(ersion)f(of)h(Theorem)f(1.1)i (whic)m(h)d(resem)m(bles)i(more)f(to)i(the)448 4299 y(standard)g Fr(N)10 b Fs(-b)s(o)s(dy)37 b(v)m(ersion,)i(w)m(e)g(shall)e(consider)g (only)g(a)i(rather)f(particular)e(class)448 4412 y(of)h Fr(H)7 b Fs(.)60 b(T)-8 b(ak)m(e)38 b Fr(X)43 b Fs(=)35 b Fr(X)1258 4379 y Fq(\003)1334 4412 y Fs(=)h Fm(R)1500 4379 y Fz(n)1590 4412 y Fs(and)g(let)h Fr(h)f Fs(:)g Fm(R)2120 4379 y Fz(n)2209 4412 y Fp(!)f Fm(R)46 b Fs(b)s(e)36 b(a)h(con)m(tin)m(uous)f(function)448 4524 y(suc)m(h)42 b(that)h Fr(c)913 4491 y Fq(\000)p Fn(1)1007 4524 y Fp(h)p Fr(x)p Fp(i)1129 4491 y Fn(2)p Fz(s)1247 4524 y Fp(\024)h Fr(h)p Fs(\()p Fr(x)p Fs(\))i Fp(\024)e Fr(c)p Fp(h)p Fr(x)p Fp(i)1858 4491 y Fn(2)p Fz(s)1973 4524 y Fs(if)d Fp(j)16 b Fr(x)f Fp(j)45 b Fr(>)f(r)s Fs(,)h(where)c Fr(s;)15 b(r)m(;)g(c)44 b Fs(are)e(strictly)448 4637 y(p)s(ositiv)m(e)25 b(constan)m(ts)j(and)d Fp(h)p Fr(x)p Fp(i)h Fs(=)f(\(1)12 b(+)g Fp(j)j Fr(x)g Fp(j)1903 4595 y Fn(2)1942 4637 y Fs(\))1977 4604 y Fn(1)p Fz(=)p Fn(2)2087 4637 y Fs(.)40 b(Then)25 b Fr(H)2461 4651 y Fn(0)2525 4637 y Fs(=)g Fr(h)p Fs(\()p Fr(P)13 b Fs(\))27 b(is)e(a)i(self-adjoin) m(t)448 4750 y(op)s(erator)36 b(whose)f(form)g(domain)f(is)g(the)h(Sob) s(olev)g(space)g Fp(H)2567 4717 y Fz(s)2604 4750 y Fs(\()p Fr(X)7 b Fs(\).)56 b(F)-8 b(or)36 b(eac)m(h)h(linear)448 4863 y(subspace)j Fr(Y)61 b Fp(\032)42 b Fr(X)47 b Fs(let)40 b Fr(V)1381 4877 y Fz(Y)1482 4863 y Fs(b)s(e)f(a)i(con)m(tin)m(uous)f (symmetric)f(sesquilinear)f(form)h(on)448 4976 y Fp(H)526 4943 y Fz(s)563 4976 y Fs(\()p Fr(X)7 b Fs(\),)31 b(iden)m(ti\014ed)e (with)g(an)h(op)s(erator)h Fp(H)1946 4943 y Fz(s)1982 4976 y Fs(\()p Fr(X)7 b Fs(\))27 b Fp(!)e(H)2355 4943 y Fq(\000)p Fz(s)2446 4976 y Fs(\()p Fr(X)7 b Fs(\),)32 b(suc)m(h)e(that:)1920 5225 y(3)p eop %%Page: 4 4 4 3 bop 448 573 a Fs(\(i\))71 b([e)681 540 y Fz(i)p Fq(h)p Fz(y)r(;P)10 b Fq(i)875 573 y Fr(;)15 b(V)968 587 y Fz(Y)1029 573 y Fs(])26 b(=)f(0)91 b Fp(8)25 b Fr(y)j Fp(2)d Fr(Y)20 b Fs(;)448 686 y(\(ii\))610 608 y Fl(\015)610 663 y(\015)660 686 y Fs([e)726 653 y Fz(i)p Fq(h)p Fz(Q;k)r Fq(i)923 686 y Fr(;)15 b(V)1016 700 y Fz(Y)1077 686 y Fs(])1102 608 y Fl(\015)1102 663 y(\015)1153 722 y Fq(H)1213 703 y Fi(s)1246 722 y Fq(!H)1377 703 y Fh(\000)p Fi(s)1487 686 y Fp(!)26 b Fs(0)f(if)g Fr(k)j Fp(2)d Fr(Y)5 b(;)30 b(k)f Fp(!)c Fs(0;)448 813 y(\(iii\))610 736 y Fl(\015)610 790 y(\015)660 813 y Fs([e)726 780 y Fz(i)p Fq(h)p Fz(Q;k)r Fq(i)943 813 y Fp(\000)20 b Fs(1])p Fr(V)1157 827 y Fz(Y)1218 736 y Fl(\015)1218 790 y(\015)1269 849 y Fq(H)1329 830 y Fi(s)1362 849 y Fq(!H)1493 830 y Fh(\000)p Fi(s)1604 813 y Fp(!)25 b Fs(0)g(if)g Fr(k)j Fp(2)d Fr(Y)2103 780 y Fq(?)2162 813 y Fr(;)41 b(k)28 b Fp(!)d Fs(0.)448 990 y(W)-8 b(e)24 b(ha)m(v)m(e)g(denoted)f(e)1175 957 y Fz(i)p Fq(h)p Fz(Q;k)r Fq(i)1395 990 y Fs(the)g(op)s(erator)g(of)g(m)m (ultiplication)d(b)m(y)i(the)h(function)f(e)3231 957 y Fz(i)p Fq(h)p Fz(x;k)r Fq(i)3412 990 y Fs(.)448 1103 y(F)-8 b(or)36 b Fr(Y)53 b Fs(=)32 b Fr(O)k Fs(=)c Fp(f)p Fs(0)p Fp(g)k Fs(w)m(e)g(tak)m(e)g Fr(V)1597 1117 y Fz(Y)1691 1103 y Fs(=)c(0.)55 b(Note)36 b(that)f(in)f(\(ii\))g(\(but)h(not)g(in)e (\(iii\)\))h(the)448 1216 y(euclidean)29 b(structure)h(of)h Fr(X)h Fs(=)25 b Fm(R)1604 1183 y Fz(n)1688 1216 y 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Fz(Y)1542 1941 y Fr(<)d Fs(1)j(and)1901 1873 y Fl(P)1997 1968 y Fz(Y)2073 1941 y Fr(\016)2113 1955 y Fz(Y)2204 1941 y Fr(<)c Fp(1)p Fs(.)49 b(Then)32 b Fr(H)k Fs(=)30 b Fr(H)2998 1955 y Fn(0)3059 1941 y Fs(+)3152 1873 y Fl(P)3247 1968 y Fz(Y)3323 1941 y Fr(V)3376 1955 y Fz(Y)448 2054 y Fs(is)j(a)i(self-adjoin)m(t,)g(b)s(ounded)c(from)j(b) s(elo)m(w)f(op)s(erator)i(on)f Fp(H)q Fs(\()p Fr(X)7 b Fs(\),)36 b Fr(H)41 b Fs(is)33 b(a\016liated)g(to)448 2167 y Fv(C)509 2134 y Fz(X)577 2167 y Fs(,)d(and)1489 2288 y Fr(\033)1541 2302 y Fn(ess)1632 2288 y Fs(\()p Fr(H)7 b Fs(\))26 b(=)1950 2202 y Fl([)1907 2398 y Fz(Y)15 b Fq(2M)2109 2288 y Fr(\033)s Fs(\()p Fr(H)2275 2302 y Fz(Y)2336 2288 y Fs(\))p Fr(:)448 2542 y Fs(Here)33 b Fr(H)741 2556 y Fz(Y)831 2542 y Fs(=)c Fr(H)1007 2556 y Fn(0)1067 2542 y Fs(+)1160 2474 y Fl(P)1256 2569 y Fz(Z)5 b Fq(\033)p Fz(Y)1439 2542 y Fr(V)1492 2556 y Fz(Z)1581 2542 y Fs(and)32 b Fp(M)h Fs(is)f(the)g(set)i(of)e(minimal)e (elemen)m(ts)j(\(for)g(the)448 2655 y(inclusion)i(relation\))h(of)i (the)f(class)g(of)h(subspaces)e(of)i(the)f(form)g Fr(Y)2786 2669 y Fn(1)2850 2655 y Fp(\\)25 b(\001)15 b(\001)g(\001)26 b(\\)e Fr(Y)3205 2670 y Fz(k)3284 2655 y Fp(6)p Fs(=)37 b(0)448 2768 y(with)29 b Fr(Y)708 2782 y Fz(i)762 2768 y Fp(2)24 b(f)p Fr(Y)36 b Fp(j)15 b Fr(V)1074 2782 y Fz(Y)1160 2768 y Fp(6)p Fs(=)25 b(0)p Fp(g)p Fs(.)589 2931 y(Theorem)k(5.2)i(can)e(also)h(b)s(e)e(used)h(in)f(order)h(to)h (pro)m(v)m(e)g(the)f(Mourre)g(estimate)h(for)448 3044 y(op)s(erators)23 b(a\016liated)f(to)i Fv(C)1374 3011 y Fz(X)1441 3044 y Fs(.)39 b(W)-8 b(e)24 b(shall)d(presen)m(t)i(here)f (only)g(the)h(simplest)e(case)j(when)448 3157 y Fr(H)524 3171 y Fn(0)597 3157 y Fs(is)32 b(the)i(\(p)s(ositiv)m(e\))f(laplacian) f(and)g(the)i Fr(V)2040 3171 y Fz(Y)2134 3157 y Fs(do)f(not)h(dep)s (end)e(on)h(the)g(pro)5 b(jection)448 3269 y(on)31 b Fr(Y)51 b Fs(of)31 b(the)g(momen)m(tum,)g(so)g Fr(V)1608 3283 y Fz(Y)1695 3269 y Fs(=)26 b(1)21 b Fp(\012)f Fr(V)2022 3237 y Fz(Y)2114 3269 y Fs(if)30 b Fp(H)q Fs(\()p Fr(X)7 b Fs(\))27 b(=)f Fp(H)q Fs(\()p Fr(Y)20 b Fs(\))g Fp(\012)h(H)q Fs(\()p Fr(Y)3071 3237 y Fq(?)3130 3269 y Fs(\).)42 b(More)448 3382 y(precisely)-8 b(,)30 b(it)g(su\016ces)g(to)h(replace)f(\(ii\))g (b)m(y)g(the)h(stronger)f(condition:)448 3520 y(\(ii)535 3487 y Fq(\003)574 3520 y Fs(\))15 b([e)690 3487 y Fz(i)p Fq(h)p Fz(Q;k)r Fq(i)886 3520 y Fr(;)g(V)979 3534 y Fz(Y)1041 3520 y Fs(])25 b(=)g(0)h(if)e Fr(k)29 b Fp(2)24 b Fr(Y)c Fs(.)448 3658 y(No)m(w)38 b Fr(H)44 b Fs(lo)s(oks)37 b(exactly)h(as)f(in)f(the)h(non-relativistic)e Fr(N)10 b Fs(-b)s(o)s(dy)36 b(problem,)i(the)f(only)448 3771 y(di\013erence)k(is)g(that)h(w)m(e)g(allo)m(w)g(an)f(in\014nite)e (lattice)j(of)g(subspaces.)74 b(Th)m(us)41 b(w)m(e)h(can)448 3884 y(de\014ne)32 b(op)s(erators)g Fr(H)1200 3851 y Fz(Y)1293 3884 y Fs(acting)h(in)e Fp(H)q Fs(\()p Fr(Y)1862 3851 y Fq(?)1921 3884 y Fs(\))i(for)f(eac)m(h)h Fr(Y)49 b Fp(\032)28 b Fr(X)39 b Fs(and)32 b(these)h(op)s(erators)448 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Fq(\003)1458 4495 y Fs(-algebra)f(generated)h(b)m(y)f(the)g(op)s(erators)g Fr(H)46 b Fs(=)40 b Fr(h)p Fs(\()p Fr(P)f Fs(+)448 4608 y Fr(k)s Fs(\))24 b(+)f Fr(V)e Fs(\()p Fr(Q)p Fs(\))33 b Fp(\021)g Fr(H)1080 4622 y Fn(0)1143 4608 y Fs(+)23 b Fr(V)d Fs(\()p Fr(Q)p Fs(\),)37 b(with)d Fr(k)j Fp(2)c Fr(X)1986 4575 y Fq(\003)2061 4608 y Fs(and)h Fr(V)54 b Fp(2)33 b(A)2516 4575 y Fq(1)2590 4608 y Fs(.)55 b(By)36 b(making)e(a)i(series)448 4721 y(expansion)30 b(for)g(large)g Fr(z)1019 4892 y Fs(\()p Fr(z)25 b Fp(\000)20 b Fr(H)7 b Fs(\))1330 4855 y Fq(\000)p Fn(1)1449 4892 y Fs(=)1546 4806 y Fl(X)1545 5002 y Fz(n)p Fq(\025)p Fn(0)1678 4892 y Fs(\()p Fr(z)25 b Fp(\000)20 b Fr(H)1947 4906 y Fn(0)1986 4892 y Fs(\))2021 4855 y Fq(\000)p Fn(1)2116 4892 y Fs([)p Fr(V)g Fs(\()p Fr(Q)p Fs(\)\()p Fr(z)26 b Fp(\000)20 b Fr(H)2626 4906 y Fn(0)2665 4892 y Fs(\))2700 4855 y Fq(\000)p Fn(1)2794 4892 y Fs(])2819 4855 y Fz(n)1920 5225 y Fs(8)p eop %%Page: 9 9 9 8 bop 448 573 a Fs(w)m(e)35 b(easily)f(get)h 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Fs(,)i(w)m(e)e(get)448 1250 y Fr(\022)s Fs(\()p Fr("H)647 1264 y Fn(0)686 1250 y Fs(\)\()p Fr(H)28 b Fp(\000)20 b Fr(z)t Fs(\))26 b Fp(2)f Fv(C)p Fs(.)40 b(Th)m(us:)448 1450 y Fr(\022)s Fs(\()p Fr("H)647 1464 y Fn(0)686 1450 y Fs(\))p Fr(V)21 b Fs(\()p Fr(Q)p Fs(\)\()p Fr(H)1048 1464 y Fn(0)1105 1450 y Fp(\000)c Fr(z)t Fs(\))1274 1413 y Fq(\000)p Fn(1)1394 1450 y Fs(=)25 b Fr(\022)s Fs(\()p Fr("H)1689 1464 y Fn(0)1728 1450 y Fs(\)\()p Fr(H)1874 1464 y Fn(0)1931 1450 y Fp(\000)17 b Fr(z)t Fs(\))h Fp(\001)g Fs(\()p Fr(H)2272 1464 y Fn(0)2328 1450 y Fp(\000)f Fr(z)t Fs(\))2497 1413 y Fq(\000)p Fn(1)2592 1450 y Fr(V)k Fs(\()p Fr(Q)p Fs(\)\()p Fr(H)2919 1464 y Fn(0)2976 1450 y Fp(\000)c Fr(z)t Fs(\))3145 1413 y Fq(\000)p Fn(1)3265 1450 y Fp(2)25 b Fv(C)p Fr(:)448 1650 y Fs(W)-8 b(e)32 b(can)e(write)g(\()p Fr(H)1115 1664 y Fn(0)1174 1650 y Fp(\000)20 b Fr(z)t Fs(\))1346 1617 y Fq(\000)p Fn(1)1466 1650 y Fs(=)25 b(lim)1689 1664 y Fz(n)p Fq(!1)1892 1650 y Fr( )1951 1664 y Fz(n)1998 1650 y Fs(\()p Fr(P)13 b Fs(\))31 b(\(norm)f(limit\))e (with)h Fr( )2961 1664 y Fz(n)3034 1650 y Fp(2)c Fr(C)3185 1664 y Fz(c)3219 1650 y Fs(\()p Fr(X)3336 1617 y Fq(\003)3377 1650 y Fs(\).)448 1763 y(Then,)46 b(for)d(eac)m(h)i Fr(s)h(>)h Fs(0,)g Fr( )1480 1777 y Fz(n)1527 1763 y Fs(\()p Fr(P)13 b Fs(\))44 b(is)e(a)i(con)m(tin)m(uous)f(map)g(from)g Fr(L)2878 1730 y Fn(2)2917 1763 y Fs(\()p Fr(X)7 b Fs(\))45 b(in)m(to)e(the)448 1876 y(Sob)s(olev)27 b(space)h Fp(H)1102 1843 y Fz(s)1139 1876 y Fs(\()p Fr(X)7 b Fs(\))28 b(and)f Fr(V)20 b Fs(\()p Fr(Q)p Fs(\))26 b Fp(2)f Fr(B)5 b Fs(\()p Fp(H)2007 1843 y Fz(s)2043 1876 y Fs(\()p Fr(X)i Fs(\)\).)42 b(Clearly)26 b(lim)2736 1890 y Fz(")p Fq(!)p Fn(0)2894 1876 y Fr(\022)s Fs(\()p Fr("H)3093 1890 y Fn(0)3132 1876 y Fs(\))f(=)g(1)j(in)448 1989 y(norm)34 b(in)g Fr(B)5 b Fs(\()p Fp(H)988 1956 y Fz(s)1024 1989 y Fs(\()p Fr(X)i Fs(\))p Fr(;)15 b(L)1278 1956 y Fn(2)1319 1989 y Fs(\()p Fr(X)7 b Fs(\)\).)56 b(Th)m(us)33 b(w)m(e)j(see)f(that)h(lim)2439 2003 y Fz(")p Fq(!)p Fn(0)2597 1989 y Fr(\022)s Fs(\()p Fr("H)2796 2003 y Fn(0)2835 1989 y Fs(\))p Fr(V)20 b Fs(\()p Fr(Q)p Fs(\))p Fr( )3144 2003 y Fz(n)3192 1989 y Fs(\()p Fr(P)13 b Fs(\))33 b(=)448 2102 y Fr(V)21 b Fs(\()p Fr(Q)p Fs(\))p Fr( )723 2116 y Fz(n)770 2102 y Fs(\()p Fr(P)13 b Fs(\))34 b(in)e(norm)g(in)g Fr(B)5 b Fs(\()p Fr(L)1574 2069 y Fn(2)1613 2102 y Fs(\()p Fr(X)i Fs(\)\))34 b(for)f(eac)m(h)h Fr(n)p Fs(.)49 b(On)32 b(the)h(other)g (hand,)g(w)m(e)h(ha)m(v)m(e)448 2138 y Fl(\015)448 2192 y(\015)499 2215 y Fr(V)20 b Fs(\()p Fr(Q)p Fs(\))p Fr( )773 2229 y Fz(n)821 2215 y Fs(\()p Fr(P)13 b Fs(\))21 b Fp(\000)f Fr(V)g Fs(\()p Fr(Q)p Fs(\)\()p Fr(H)1400 2229 y Fn(0)1460 2215 y Fp(\000)g Fr(z)t Fs(\))1632 2182 y Fq(\000)p Fn(1)1727 2138 y Fl(\015)1727 2192 y(\015)1826 2215 y Fp(!)49 b Fs(0)c(as)g Fr(n)j Fp(!)h(1)p Fs(.)83 b(It)45 b(follo)m(ws)e(then)h (that)448 2328 y(lim)575 2342 y Fz(")p Fq(!)p Fn(0)732 2328 y Fr(\022)s Fs(\()p Fr("H)931 2342 y Fn(0)970 2328 y Fs(\))p Fr(V)21 b Fs(\()p Fr(Q)p Fs(\)\()p Fr(H)1332 2342 y Fn(0)1392 2328 y Fp(\000)f Fr(z)t Fs(\))1564 2295 y Fq(\000)p Fn(1)1684 2328 y Fs(=)25 b Fr(V)20 b Fs(\()p Fr(Q)p Fs(\)\()p Fr(H)2106 2342 y Fn(0)2166 2328 y Fp(\000)g Fr(z)t Fs(\))2338 2295 y Fq(\000)p Fn(1)2464 2328 y Fs(in)29 b(norm)g(in)g Fr(B)5 b Fs(\()p Fr(L)3084 2295 y Fn(2)3124 2328 y Fs(\()p Fr(X)i Fs(\)\).)589 2441 y(This)29 b(argumen)m(t)i(pro)m (v)m(es)g(that)1023 2641 y Fr(V)20 b Fs(\()p Fr(Q)p Fs(\)\()p Fr(h)p Fs(\()p Fr(P)35 b Fs(+)20 b Fr(k)s Fs(\))g Fp(\000)g Fr(z)t Fs(\))1821 2603 y Fq(\000)p Fn(1)1941 2641 y Fs(=)25 b Fr(V)c Fs(\()p Fr(Q)p Fs(\)\()p Fr(H)2364 2655 y Fn(0)2424 2641 y Fp(\000)f Fr(z)t Fs(\))2596 2603 y Fq(\000)p Fn(1)2716 2641 y Fp(2)25 b Fv(C)448 2841 y Fs(for)30 b(eac)m(h)i Fr(k)c Fp(2)d Fr(X)1035 2808 y Fq(\003)1075 2841 y Fs(.)41 b(This)28 b(clearly)i(implies)e Fr(V)20 b Fs(\()p Fr(Q)p Fs(\))p Fr(\030)t Fs(\()p Fr(H)2318 2855 y Fn(0)2358 2841 y Fs(\))25 b Fp(2)g Fv(C)31 b Fs(for)f Fr(\030)f Fp(2)c Fr(C)2955 2855 y Fz(c)2989 2841 y Fs(\()p Fm(R)t Fs(\).)589 2953 y(The)39 b(set)h(of)f Fr( )k Fp(2)d Fr(C)1316 2967 y Fn(0)1355 2953 y Fs(\()p Fr(X)1472 2920 y Fq(\003)1513 2953 y Fs(\))f(suc)m(h)g(that)h Fr(V)20 b Fs(\()p Fr(Q)p Fs(\))p Fr( )s Fs(\()p Fr(P)13 b Fs(\))41 b Fp(2)f Fv(C)f Fs(is)f(norm)h(closed)g(and)448 3066 y(con)m(tains)34 b(all)f(the)h(functions)e(of)i(the)g(form)g Fr( )s Fs(\()p Fr(P)13 b Fs(\))32 b(=)e Fr(\030)t Fs(\()p Fr(h)p Fs(\()p Fr(P)37 b Fs(+)22 b Fr(k)s Fs(\)\))35 b(with)d Fr(\030)j Fp(2)c Fr(C)3266 3080 y Fz(c)3300 3066 y Fs(\()p Fm(R)t Fs(\))448 3179 y(and)44 b Fr(k)51 b Fp(2)d Fr(X)928 3146 y Fq(\003)968 3179 y Fs(.)81 b(The)44 b(family)e(consisting)h(of)h(suc) m(h)g(functions)f(is)g(a)h Fp(\003)p Fs(-subalgebra)448 3292 y(of)c Fr(C)626 3306 y Fn(0)665 3292 y Fs(\()p Fr(X)782 3259 y Fq(\003)822 3292 y Fs(\))g(whic)m(h)e(separates)i(the)g(p)s(oin) m(ts)e(of)h Fr(X)2212 3259 y Fq(\003)2291 3292 y Fs(\(b)s(ecause)h Fp(j)15 b Fr(h)p Fs(\()p Fr(p)21 b Fs(+)f Fr(k)s Fs(\))15 b Fp(j)41 b(!)f(1)f Fs(if)448 3405 y Fr(k)29 b Fp(!)c(1)p Fs(\).)39 b(By)26 b(the)g(Stone-W)-8 b(eierstrass)26 b(theorem,)h(w)m(e)f(see)g(that)h(this)d(family)g(is)g(dense)448 3518 y(in)g Fr(C)614 3532 y Fn(0)653 3518 y Fs(\()p Fr(X)770 3485 y Fq(\003)811 3518 y Fs(\).)39 b(So)25 b(w)m(e)g(ha)m(v)m(e)h Fr(V)21 b Fs(\()p Fr(Q)p Fs(\))p Fr( )s Fs(\()p Fr(P)13 b Fs(\))27 b Fp(2)d Fv(C)i Fp(8)p Fr( )i Fp(2)d Fr(C)2270 3532 y Fn(0)2309 3518 y Fs(\()p Fr(X)2426 3485 y Fq(\003)2467 3518 y Fs(\).)39 b(Here)25 b Fr(V)45 b Fs(is)24 b(an)h(arbitrary)448 3631 y(function)32 b(in)h Fp(A)990 3598 y Fq(1)1064 3631 y Fs(.)50 b(Since)32 b Fp(A)1452 3598 y Fq(1)1560 3631 y Fs(is)h(dense)g(in)f Fp(A)p Fs(,)i(w)m(e)g(\014nally)d(obtain)i Fr(')p Fs(\()p Fr(Q)p Fs(\))p Fr( )s Fs(\()p Fr(P)13 b Fs(\))33 b Fp(2)d Fv(C)448 3744 y Fs(for)g(all)g Fr(')25 b Fp(2)g(A)p Fr(;)15 b( )29 b Fp(2)c Fr(C)1236 3758 y Fn(0)1275 3744 y Fs(\()p Fr(X)1392 3711 y Fq(\003)1433 3744 y Fs(\).)p 3371 3736 67 67 v 448 3928 a Fj(Corollary)35 b(2.4)46 b Fo(L)-5 b(et)33 b Fr(h)26 b Fs(:)f Fr(X)1459 3895 y Fq(\003)1524 3928 y Fp(!)h Fm(R)41 b Fo(b)-5 b(e)32 b(an)h(el)5 b(liptic)33 b(p)-5 b(olynomial)35 b(of)e(or)-5 b(der)34 b Fr(m)p Fo(.)42 b(Then)448 4041 y(the)32 b Fr(C)670 4008 y Fq(\003)709 4041 y Fo(-algebr)-5 b(a)32 b(of)f(op)-5 b(er)g(ators)34 b(on)e Fp(H)q Fs(\()p Fr(X)7 b Fs(\))32 b Fo(gener)-5 b(ate)g(d)32 b(by)f(the)h(self-adjoint)g(op)-5 b(er)g(ators)448 4154 y Fr(H)33 b Fs(=)24 b Fr(h)p Fs(\()p Fr(P)13 b Fs(\))18 b(+)e Fr(W)d Fo(,)31 b(wher)-5 b(e)31 b Fr(W)44 b Fo(is)31 b(a)g(symmetric)h(di\013er)-5 b(ential)32 b(op)-5 b(er)g(ator)34 b(of)d(or)-5 b(der)32 b Fr(<)25 b(m)448 4267 y Fo(with)34 b(c)-5 b(o)g(e\016cients)33 b(in)g Fp(A)1294 4234 y Fq(1)1368 4267 y Fo(,)g(is)f(e)-5 b(qual)33 b(to)g Fp(A)20 b Fk(o)g Fr(X)7 b Fo(.)448 4428 y Fj(Pro)s(of.)48 b Fs(If)30 b Fr(V)45 b Fp(2)25 b(A)1126 4395 y Fq(1)1200 4428 y Fs(,)31 b(and)f(if)f(w)m(e)i(iden)m(tify)e Fr(X)2063 4395 y Fq(\003)2128 4428 y Fs(=)c Fm(R)2284 4395 y Fz(n)2337 4428 y Fs(,)30 b(w)m(e)h(ha)m(v)m(e)647 4666 y Fr(h)p Fs(\()p Fr(P)j Fs(+)20 b Fr(k)s Fs(\))g(+)g Fr(V)g Fs(\()p Fr(Q)p Fs(\))26 b(=)f Fr(h)p Fs(\()p Fr(P)13 b Fs(\))22 b(+)1789 4579 y Fl(X)1755 4781 y Fq(j)12 b Fz(\013)g Fq(j\025)p Fn(1)1979 4604 y Fr(k)2029 4571 y Fz(\013)p 1979 4645 100 4 v 1987 4728 a Fr(\013)p Fs(!)2088 4666 y Fr(h)2140 4628 y Fn(\()p Fz(\013)p Fn(\))2245 4666 y Fs(\()p Fr(P)h Fs(\))21 b(+)f Fr(V)g Fs(\()p Fr(Q)p Fs(\))26 b Fp(\021)f Fr(h)p Fs(\()p Fr(P)13 b Fs(\))21 b(+)f Fr(W)448 4976 y Fs(so)31 b(w)m(e)g(ma)m(y)g(use)f(the)g (preceding)g(theorem.)p 3371 4968 67 67 v 1920 5225 a(9)p eop %%Page: 10 10 10 9 bop 448 573 a Ft(3)135 b(Graded)45 b Fg(C)1225 529 y Ff(\003)1270 573 y Ft(-Algebras)448 776 y Fs(In)31 b(this)g(pap)s(er)g(w)m(e)h(are)g(in)m(terested)g(in)e Fr(C)1885 743 y Fq(\003)1924 776 y Fs(-algebras)i(whic)m(h)f(are)h (graded)g(b)m(y)f(a)i(semi-)448 889 y(lattice)d Fp(L)p Fs(.)40 b(The)29 b(case)h(when)e Fp(L)h Fs(is)g(\014nite)f(is)g(presen) m(ted)h(in)f(Section)h(8.4)i(from)d([ABG].)448 1002 y(Belo)m(w)35 b(w)m(e)g(extend)f(the)h(formalism)d(to)j(the)g(case)g(of)f(in\014nite) e Fp(L)p Fs(.)52 b(Note)36 b(that)f(in)e(the)448 1115 y(presen)m(t)27 b(con)m(text)h(it)d(is)h(con)m(v)m(enien)m(t)h(to)g(in) m(terc)m(hange)g(the)f(r^)-45 b(oles)26 b(of)g(the)h(lo)m(w)m(er)f(and) g(up-)448 1227 y(p)s(er)i(b)s(ounds)e(in)i(the)g(de\014nition)f(of)h (the)h(grading,)g(whic)m(h)e(explains)g(some)i(di\013erences)448 1340 y(in)g(notations)i(with)e(resp)s(ect)h(to)h([ABG].)589 1503 y(Let)k Fp(L)e Fs(b)s(e)g(an)h(arbitrary)f Fo(semilattic)-5 b(e)p Fs(,)36 b(i.e.)50 b Fp(L)34 b Fs(is)f(a)h(partially)e(ordered)h (set)h(suc)m(h)448 1616 y(that)25 b(the)f(lo)m(w)m(er)h(b)s(ound)d Fr(a)8 b Fp(^)g Fr(b)23 b Fs(exists)h(for)g(all)f Fr(a;)15 b(b)26 b Fp(2)f(L)p Fs(.)38 b(If)24 b Fp(L)g Fs(has)g(a)g(least)g(or)h (a)f(greatest)448 1729 y(elemen)m(t,)33 b(w)m(e)f(denote)g(it)f(min)14 b Fp(L)31 b Fs(or)g(max)16 b Fp(L)31 b Fs(resp)s(ectiv)m(ely)-8 b(.)44 b(W)-8 b(e)33 b(denote)f(b)m(y)f Fm(F)5 b Fs(\()p Fp(L)p Fs(\))38 b(the)448 1842 y(family)c(of)h(\014nite)e(subsets)i Fp(F)42 b(\032)32 b(L)j Fs(suc)m(h)f(that)i Fr(a)23 b Fp(^)g Fr(b)32 b Fp(2)h(F)44 b Fs(if)34 b Fr(a;)15 b(b)33 b Fp(2)f(F)44 b Fs(\(the)36 b(empt)m(y)448 1955 y(set)d(b)s(elongs)d (to)j Fm(F)5 b Fs(\()p Fp(L)p Fs(\)\).)51 b(So)32 b(eac)m(h)h Fp(F)k(2)27 b Fm(F)5 b Fs(\()p Fp(L)p Fs(\))38 b(is)31 b(a)h(\014nite)f(semilattice)g(for)h(the)g(order)448 2068 y(relation)d(induced)f(b)m(y)h Fp(L)p Fs(,)h(in)e(particular)g(it) i(has)f(a)h(least)g(elemen)m(t)g(min)14 b Fp(F)9 b Fs(.)41 b(W)-8 b(e)30 b(equip)448 2181 y Fm(F)5 b Fs(\()p Fp(L)p Fs(\))44 b(with)37 b(the)h(order)f(relation)g(giv)m(en)h(b)m(y)g (inclusion.)60 b(Since)37 b Fp(F)2766 2195 y Fn(1)2831 2181 y Fp(\\)24 b(F)2981 2195 y Fn(2)3059 2181 y Fp(2)37 b Fm(F)5 b Fs(\()p Fp(L)p Fs(\))44 b(if)448 2293 y Fp(F)513 2307 y Fn(1)553 2293 y Fr(;)15 b Fp(F)658 2307 y Fn(2)724 2293 y Fp(2)24 b Fm(F)5 b Fs(\()q Fp(L)p Fs(\),)34 b(the)28 b(set)h Fm(F)t Fs(\()q Fp(L)p Fs(\))34 b(b)s(ecomes)28 b(a)g(semilattice.)40 b(Note)29 b(that)f(for)g(eac)m(h)h(\014nite)448 2406 y(part)g Fr(F)38 b Fp(\032)25 b(L)k Fs(the)g(set)g(of)g(elemen)m (ts)g(of)g(the)g(form)g Fr(a)2214 2420 y Fn(1)2270 2406 y Fp(^)17 b(\001)g(\001)g(\001)h(^)e Fr(a)2600 2420 y Fz(n)2676 2406 y Fs(with)28 b Fr(a)2930 2420 y Fn(1)2969 2406 y Fr(;)15 b Fp(\001)j(\001)f(\001)p Fr(;)e(a)3207 2420 y Fz(n)3280 2406 y Fp(2)25 b Fr(F)448 2519 y Fs(b)s(elongs)30 b(to)h Fm(F)5 b Fs(\()p Fp(L)p Fs(\))36 b(and)30 b(con)m(tains)h Fr(F)13 b Fs(.)589 2682 y(A)32 b Fp(L)p Fo(-gr)-5 b(ade)g(d)34 b Fr(C)1141 2649 y Fq(\003)1180 2682 y Fo(-algebr)-5 b(a)32 b Fs(is)e(a)h Fr(C)1759 2649 y Fq(\003)1798 2682 y Fs(-algebra)h Fv(C)f Fs(equipp)s(ed)d(with)i(a)h(linearly)e(inde-)448 2795 y(p)s(enden)m(t)h(family)f Fp(f)p Fv(C)p Fs(\()p Fr(a)p Fs(\))p Fp(g)1344 2809 y Fz(a)p Fq(2L)1513 2795 y Fs(of)h Fr(C)1688 2762 y Fq(\003)1727 2795 y Fs(-subalgebras)g(suc)m (h)g(that:)448 2908 y(\(i\))h Fv(C)p Fs(\()p Fr(a)p Fs(\))21 b Fp(\001)f Fv(C)p Fs(\()p Fr(b)p Fs(\))26 b Fp(\032)f Fv(C)p Fs(\()p Fr(a)c Fp(^)e Fr(b)p Fs(\))31 b(for)f(all)g Fr(a;)15 b(b)25 b Fp(2)g(L)p Fs(;)448 3021 y(\(ii\))30 b(if)f Fp(F)35 b(2)25 b Fm(F)t Fs(\()q Fp(L)p Fs(\))36 b(then)30 b Fv(C)p Fs(\()p Fp(F)9 b Fs(\))27 b(:=)1648 2953 y Fl(P)1744 3048 y Fz(a)p Fq(2F)1905 3021 y Fv(C)p Fs(\()p Fr(a)p Fs(\))k(is)e(a)i(closed)f(subspace)g(of)g Fv(C)p Fs(;)448 3134 y(\(iii\))625 3065 y Fl(S)701 3160 y Fq(F)7 b(2)p Fe(F)p Fn(\()p Fq(L)p Fn(\))968 3134 y Fv(C)p Fs(\()p Fp(F)i Fs(\))27 b Fp(\021)1296 3065 y Fl(P)1391 3160 y Fz(a)p Fq(2L)1544 3134 y Fv(C)p Fs(\()p Fr(a)p Fs(\))k(is)e(dense)h(in)f Fv(C)p Fs(.)589 3296 y(It)39 b(is)f(clear)g(that)h(for)g(eac)m(h)h Fp(F)48 b(2)38 b Fm(F)5 b Fs(\()p Fp(L)p Fs(\))45 b(the)39 b(space)g Fv(C)p Fs(\()p Fp(F)9 b Fs(\))39 b(is)f(a)h Fr(C)2943 3263 y Fq(\003)2982 3296 y Fs(-subalgebra)448 3409 y(of)h Fv(C)g 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Fj(Theorem)34 b(3.1)46 b Fo(F)-7 b(or)34 b(al)5 b(l)32 b Fr(a)26 b Fp(2)f(L)31 b Fo(one)i(has)g Fv(C)26 b Fs(=)f Fv(C)2202 587 y Fz(a)2263 573 y Fs(+)19 b Fv(J)2408 587 y Fz(a)2482 573 y Fo(and)33 b Fv(C)2719 587 y Fz(a)2780 573 y Fp(\\)19 b Fv(J)2915 587 y Fz(a)2982 573 y Fs(=)25 b Fp(f)p Fs(0)p Fp(g)p Fo(.)43 b(The)448 686 y(pr)-5 b(oje)g(ction)37 b Fp(P)935 700 y Fz(a)1007 686 y Fs(:)30 b Fv(C)g Fp(!)g Fv(C)1335 700 y Fz(a)1411 686 y Fo(determine)-5 b(d)37 b(by)e(this)h(line)-5 b(ar)36 b(dir)-5 b(e)g(ct)36 b(sum)g(de)-5 b(c)g(omp)g(osition)448 799 y(is)33 b(a)g(morphism,)i(in)d(p)-5 b(articular)35 b Fp(k)q(P)1713 813 y Fz(a)1754 799 y Fp(k)26 b Fs(=)f(1)p Fo(.)448 953 y Fj(Pro)s(of.)48 b Fs(W)-8 b(e)32 b(clearly)f(ha)m(v)m(e) h Fv(C)p Fs(\()p Fp(L)p Fs(\))c(=)e Fv(C)p Fs(\()p Fp(L)1915 967 y Fz(a)1957 953 y Fs(\))21 b(+)g Fv(C)p Fs(\()p Fp(L)2264 920 y Fq(0)2264 975 y Fz(a)2305 953 y Fs(\))32 b(as)g(a)f(linear)f (direct)h(sum.)43 b(Let)448 1065 y Fp(P)518 1033 y Fq(\016)511 1088 y 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b(that)g Fp(L)f Fs(is)g Fo(atomic)h Fs(if)f(eac)m(h)h Fr(b)c Fp(6)p Fs(=)g(min)14 b Fp(L)33 b Fs(is)f(minorated)h(b)m(y)g(an) h(atom.)50 b(Then)33 b(w)m(e)448 1024 y(can)e(asso)s(ciate)g(to)g Fv(C)g Fs(a)g(second)f Fr(C)1641 992 y Fq(\003)1680 1024 y Fs(-algebra,)h(namely)1701 1195 y Fl(e)1696 1225 y Fv(C)25 b Fs(:=)1919 1139 y Fl(M)1903 1335 y Fz(a)p Fq(2M)2087 1225 y Fv(C)2148 1239 y Fz(a)448 1225 y Fs(\(3.3\))448 1500 y(where)38 b(the)h(direct)f(sum)f(is)h(in)f(the)i Fr(C)1801 1467 y Fq(\003)1839 1500 y Fs(-algebra)g(sense.)65 b(Observ)m(e)38 b(that)i(there)e(is)g(a)448 1621 y(natural)30 b(morphism)e Fp(P)33 b Fs(:)25 b Fv(C)g Fp(!)1546 1592 y Fl(e)1541 1621 y Fv(C)p Fs(,)30 b(namely)1544 1813 y Fp(P)7 b Fs([)p Fr(T)13 b Fs(])26 b(=)f(\()p Fp(P)1950 1827 y Fz(a)1992 1813 y Fs([)p Fr(T)13 b Fs(]\))2143 1827 y Fz(a)p Fq(2M)2316 1813 y Fr(:)448 2005 y Fj(Theorem)34 b(3.2)46 b Fo(Assume)e(that)i(the)f(semilattic)-5 b(e)45 b Fp(L)f Fo(has)h(a)g(le)-5 b(ast)45 b(element)g(and)g(is)448 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4061 y(from)26 b([ABG],)i(the)e(map)g Fr(U)35 b Fp(7!)25 b Fs(\()p Fp(P)1626 4076 y Fz(b)1661 4061 y Fs([)p Fr(U)10 b Fs(]\))1818 4076 y Fz(b)p Fq(2N)1968 4061 y Fs(,)27 b(sending)d Fv(C)p Fs(\()p Fp(F)9 b Fs(\))28 b(in)m(to)2757 3993 y Fl(L)2858 4088 y Fz(b)p Fq(2N)3022 4061 y Fv(C)p Fs(\()p Fp(F)3183 4076 y Fz(b)3218 4061 y Fs(\),)f(has)448 4174 y Fv(C)p Fs(\(min)14 b Fp(F)9 b Fs(\))26 b(=)f Fv(C)p Fs(\(min)14 b Fp(L)p Fs(\))21 b(as)f(its)g(k)m(ernel.)37 b(The)19 b(map)h Fv(C)p Fs(\()p Fp(F)9 b Fs(\))p Fr(=)p Fv(C)p Fs(\(min)16 b Fp(L)p Fs(\))25 b Fp(!)2964 4106 y Fl(L)3065 4201 y Fz(b)p Fq(2N)3230 4174 y Fv(C)p Fs(\()p Fp(F)3391 4189 y Fz(b)3426 4174 y Fs(\))448 4287 y(will)h(b)s(e)i(an)h (isometry)f(and)g(since)f Fp(k)q(P)1739 4302 y Fz(b)1773 4287 y Fs([)p Fr(S)5 b Fs(])p Fp(k)27 b(\024)e Fr(")j Fs(for)h(eac)m(h)h Fr(b)25 b Fp(2)g(N)13 b Fs(,)29 b(the)f(image)h(of)g Fr(S)k Fs(in)448 4400 y(the)22 b(quotien)m(t)g(space)h Fv(C)p Fs(\()p Fp(F)9 b Fs(\))p Fr(=)p Fv(C)p Fs(\(min)15 b 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Fp(2)25 b Fv(C)p Fs(\(min)14 b Fp(L)p Fs(\).)p 3371 565 67 67 v 589 749 a(The)40 b(preceding)g(theorem)h(giv)m(es)g(us)f(a) h(canonical)f(em)m(b)s(edding)2900 719 y Fl(b)2894 749 y Fv(C)j Fp(\032)3116 719 y Fl(e)3111 749 y Fv(C)p Fs(,)g(more)448 862 y(precisely)1477 983 y Fv(C)p Fr(=)p Fv(C)p Fs(\(min)15 b Fp(L)p Fs(\))25 b Fr(,)-15 b Fp(!)2112 897 y Fl(M)2096 1094 y Fz(a)p Fq(2M)2280 983 y Fv(C)2341 997 y Fz(a)2383 983 y Fr(:)-1960 b Fs(\(3.4\))448 1233 y(Although)33 b(easy)h(to)g(pro)m(v)m(e,)i(this)c(result)g(is)h(imp)s(ortan)m(t:)46 b(it)33 b(allo)m(ws)g(one)h(to)g(compute)448 1346 y(the)27 b(essen)m(tial)f(sp)s(ectrum)g(and)f(to)j(pro)m(v)m(e)f(the)g(Mourre)f (estimate)h(under)e(v)m(ery)i(general)448 1458 y(assumptions.)56 b(The)36 b(range)g(of)g(the)g(map)g(\(3.4\))i(can)e(b)s(e)g(explicitly) d(describ)s(ed,)j(but)448 1571 y(this)30 b(is)f(irrelev)-5 b(an)m(t)30 b(for)g(our)g(purp)s(oses.)448 1858 y Ft(4)135 b Fg(C)743 1814 y Ff(\003)788 1858 y Ft(-Algebras)45 b(Asso)t(ciated)h(to)f(Subspaces)448 2139 y Fj(4.1.)39 b Fs(Let)23 b Fr(X)31 b Fs(b)s(e)22 b(a)h(\014nite)f(dimensional)f (real)h(v)m(ector)j(space)e(and)g Fr(Y)43 b Fs(a)23 b(linear)e (subspace.)448 2252 y(W)-8 b(e)32 b(denote)f Fr(\031)950 2266 y Fz(Y)1036 2252 y Fs(=)25 b Fr(\031)1187 2219 y Fz(X)1184 2279 y(Y)1285 2252 y Fs(the)30 b(canonical)h(surjection)e(of) i Fr(X)38 b Fs(on)m(to)31 b(the)g(quotien)m(t)f(v)m(ector)448 2365 y(space)35 b Fr(X=)5 b(Y)55 b Fs(and)34 b Fr(Y)1182 2332 y Fq(?)1275 2365 y Fs(the)h(set)g(of)g Fr(x)1742 2332 y Fq(\003)1813 2365 y Fp(2)d Fr(X)1988 2332 y Fq(\003)2062 2365 y Fs(suc)m(h)i(that)h Fp(h)p Fr(y)s(;)15 b(x)2647 2332 y Fq(\003)2687 2365 y Fp(i)32 b Fs(=)g(0)j Fp(8)p Fr(y)f Fp(2)e Fr(Y)20 b Fs(.)53 b(W)-8 b(e)448 2478 y(ha)m(v)m(e)32 b(canonical)e(iden)m(ti\014cations)f(\()p Fr(X=)5 b(Y)20 b Fs(\))1909 2445 y Fq(\003)1974 2478 y Fs(=)25 b Fr(Y)2143 2445 y Fq(?)2233 2478 y Fs(and)30 b Fr(X)2492 2445 y Fq(\003)2531 2478 y Fr(=)5 b(Y)2655 2445 y Fq(?)2739 2478 y Fs(=)25 b Fr(Y)2908 2445 y Fq(\003)2948 2478 y Fs(.)589 2640 y(W)-8 b(e)43 b(shall)c(em)m(b)s(ed)h Fr(C)1343 2654 y Fn(0)1383 2640 y Fs(\()p Fr(X=)5 b(Y)20 b Fs(\))44 b Fp(\032)e Fr(C)1873 2655 y Fz(b)1908 2640 y Fs(\()p Fr(X)7 b Fs(\))42 b(with)e(the)h(help)e(of)i(the)h(map)e Fr(')k Fp(7\000)-16 b(!)448 2753 y Fr(')21 b Fp(\016)g Fr(\031)646 2767 y Fz(Y)706 2753 y Fs(.)41 b(Since)29 b Fr(C)1074 2768 y Fz(b)1109 2753 y Fs(\()p Fr(X)7 b Fs(\))26 b Fp(\032)f Fv(B)p Fs(\()p Fr(X)7 b Fs(\),)32 b(w)m(e)f(shall)d(ha)m(v)m(e)1090 2958 y Fr(C)1155 2972 y Fn(0)1194 2958 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\))k Fp(\032)g Fr(C)1649 2973 y Fz(bu)1724 2958 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\))26 b Fp(\032)f Fr(C)2180 2973 y Fz(bu)2255 2958 y Fs(\()p Fr(X)7 b Fs(\))26 b Fp(\032)f Fv(B)p Fs(\()p Fr(X)7 b Fs(\))p Fr(:)-2346 b Fs(\(4.1\))448 3162 y(F)-8 b(or)40 b Fr(')f Fp(2)g Fr(C)883 3177 y Fz(b)917 3162 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\))39 b(w)m(e)g(shall)e(denote)i Fr(')p Fs(\()p Fr(Q)2055 3176 y Fz(Y)2117 3162 y Fs(\))g(=)g(\()p Fr(')26 b Fp(\016)g Fr(\031)2544 3176 y Fz(Y)2605 3162 y Fs(\)\()p Fr(Q)p Fs(\))40 b(the)e(op)s(erator)h(in)448 3275 y Fv(B)p Fs(\()p Fr(X)7 b Fs(\))26 b(asso)s(ciated)f(to)h(it.)38 b(Sometimes)25 b(it)f(is)g(imp)s(ortan)m(t)g(to)h(sp)s(ecify)f(in)f (the)i(notations)448 3388 y(the)31 b(space)g Fr(X)7 b Fs(;)31 b(then)f(w)m(e)h(set)g Fr(')p Fs(\()p Fr(Q)1635 3402 y Fz(Y)1696 3388 y Fs(\))26 b(=)f Fr(')p Fs(\()p Fr(Q)2019 3355 y Fz(X)2019 3415 y(Y)2087 3388 y Fs(\).)589 3550 y(The)30 b(relation)g Fr(Y)1183 3517 y Fq(\003)1248 3550 y Fs(=)25 b Fr(X)1426 3517 y Fq(\003)1465 3550 y Fr(=)5 b(Y)1589 3517 y Fq(?)1678 3550 y Fs(implies)1103 3755 y Fr(C)1168 3770 y Fz(b)1203 3755 y Fs(\()p Fr(Y)1311 3717 y Fq(\003)1350 3755 y Fs(\))26 b(=)f Fr(C)1572 3770 y Fz(b)1606 3755 y Fs(\()p Fr(X)1723 3717 y Fq(\003)1763 3755 y Fr(=)5 b(Y)1887 3717 y Fq(?)1946 3755 y Fs(\))26 b Fp(\032)f Fr(C)2168 3770 y Fz(b)2202 3755 y Fs(\()p Fr(X)2319 3717 y Fq(\003)2359 3755 y Fs(\))h Fp(\032)f Fv(B)p Fs(\()p Fr(X)7 b Fs(\))p Fr(:)-2333 b Fs(\(4.2\))448 3959 y(F)-8 b(or)24 b Fr( )k Fp(2)d Fr(C)842 3974 y Fz(b)876 3959 y Fs(\()p Fr(Y)985 3926 y Fq(\003)1024 3959 y Fs(\))e(w)m(e)g (denote)g Fr( )s Fs(\()p Fr(P)1648 3973 y Fz(Y)1710 3959 y Fs(\))g(or)f Fr( )s Fs(\()p Fr(P)2039 3926 y Fz(X)2026 3986 y(Y)2107 3959 y Fs(\))h(the)g(op)s(erator)g(in)e Fv(B)p Fs(\()p Fr(X)7 b Fs(\))24 b(asso)s(ciated)448 4072 y(to)i(it;)g(w)m(e)f(ha)m(v)m(e)h Fr( )s Fs(\()p Fr(P)1166 4039 y Fz(X)1153 4099 y(Y)1234 4072 y Fs(\))g(=)f(\010)1457 4086 y Fz(X)1520 4067 y Fh(\003)1560 4072 y Fs([)p Fr( )s Fs(\()p Fr(Q)1754 4039 y Fz(X)1817 4015 y Fh(\003)1754 4107 y Fz(Y)1811 4088 y Fh(?)1868 4072 y Fs(].)39 b(Observ)m(e,)26 b(in)d(particular,)h(that)i(the)f(group)448 4185 y Fr(C)520 4152 y Fq(\003)559 4185 y Fs(-algebra)31 b Fr(C)973 4199 y Fn(0)1012 4185 y Fs(\()p Fr(Y)1121 4152 y Fq(\003)1160 4185 y Fs(\))g(of)f(the)h(additiv)m(e)e(group)h Fr(Y)51 b Fs(is)29 b(em)m(b)s(edded)g(in)g Fv(B)p Fs(\()p Fr(X)7 b Fs(\).)589 4347 y(Let)36 b Fm(G)19 b Fs(\()q Fr(X)7 b Fs(\))42 b(b)s(e)34 b(the)i(grassmannian)e(of)h Fr(X)7 b Fs(,)38 b(i.e.)55 b(the)36 b(lattice)f(of)h(all)e(v)m(ector)j(sub-) 448 4460 y(spaces)i(of)g Fr(X)45 b Fs(with)38 b(inclusion)d(as)k(order) e(relation.)64 b(Note)40 b(that)f(for)f Fr(Y)5 b(;)15 b(Z)46 b Fp(2)38 b Fm(G)19 b Fs(\()p Fr(X)8 b Fs(\))448 4573 y(one)41 b(has)f Fr(Y)47 b Fp(^)27 b Fr(Z)48 b Fs(=)42 b Fr(Y)47 b Fp(\\)26 b Fr(Z)47 b Fs(and)40 b Fr(Y)47 b Fp(_)26 b Fr(Z)49 b Fs(=)42 b Fr(Y)47 b Fs(+)26 b Fr(Z)7 b Fs(.)71 b(F)-8 b(or)41 b(eac)m(h)g Fr(Y)62 b Fp(2)42 b Fm(G)19 b Fs(\()p Fr(X)8 b Fs(\))47 b(w)m(e)448 4686 y(ha)m(v)m(e)32 b(a)f Fr(C)806 4653 y Fq(\003)845 4686 y Fs(-subalgebra)f Fr(C)1396 4700 y Fn(0)1435 4686 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\))31 b(of)g Fr(C)1904 4701 y Fz(b)1938 4686 y Fs(\()p Fr(X)7 b Fs(\))32 b(as)e(explained)f(ab)s(o) m(v)m(e.)43 b(In)30 b(particular)448 4799 y Fr(C)513 4813 y Fn(0)553 4799 y Fs(\()p Fr(X=O)s Fs(\))c(=)f Fr(C)1002 4813 y Fn(0)1041 4799 y Fs(\()p Fr(X)7 b Fs(\))30 b(and)d Fr(C)1462 4813 y Fn(0)1502 4799 y Fs(\()p Fr(X=X)7 b Fs(\))26 b(=)f Fm(C)18 b Fs(.)46 b(Note)30 b(that)f(eac)m(h)g Fr(C)2707 4813 y Fn(0)2747 4799 y Fs(\()p Fr(X=)5 b(Y)20 b Fs(\))29 b(is)e(transla-)448 4912 y(tion)j(in)m(v)-5 b(arian)m(t,)30 b(i.e.)41 b(it)29 b(is)h(stable)g(under)f(all)g(the)i (automorphisms)d Fr(\034)2862 4926 y Fz(x)2906 4912 y Fs(,)i Fr(x)c Fp(2)f Fr(X)7 b Fs(.)1897 5225 y(13)p eop %%Page: 14 14 14 13 bop 589 573 a Fs(If)30 b Fp(F)35 b(\032)25 b Fm(G)19 b Fs(\()p Fr(X)8 b Fs(\))36 b(is)30 b(a)h(family)d(of)j(v)m(ector)h (subspaces)e(of)g Fr(X)38 b Fs(then)30 b(w)m(e)h(set)1453 784 y Fr(C)1525 747 y Fz(X)1518 807 y Fn(0)1592 784 y Fs(\()p Fp(F)9 b Fs(\))26 b(=)1873 698 y Fl(X)1858 895 y Fz(Y)16 b Fq(2F)2034 784 y Fr(C)2099 798 y Fn(0)2138 784 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\))p Fr(:)-1984 b Fs(\(4.3\))448 1070 y(This)33 b(is)g(the)i(linear)e(subspace)h(of)g Fr(C)1729 1085 y Fz(bu)1804 1070 y Fs(\()p Fr(X)7 b Fs(\))36 b(generated)f(b)m(y)2541 1001 y Fl(S)2616 1097 y Fz(Y)16 b Fq(2F)2796 1070 y Fr(C)2861 1084 y Fn(0)2901 1070 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\).)53 b(Note)448 1183 y(that)31 b Fr(C)717 1150 y Fz(X)710 1207 y Fn(0)784 1183 y Fs(\()p Fp(;)p Fs(\))c(=)e Fp(f)p Fs(0)p Fp(g)31 b Fs(and)f Fr(C)1437 1150 y Fz(X)1430 1207 y Fn(0)1504 1183 y Fs(\()p Fr(Y)20 b Fs(\))26 b Fp(\021)f Fr(C)1841 1150 y Fz(X)1834 1207 y Fn(0)1908 1183 y Fs(\()p Fp(f)p Fr(Y)20 b Fp(g)p Fs(\))26 b(=)f Fr(C)2328 1197 y Fn(0)2368 1183 y Fs(\()p Fr(X=)5 b(Y)20 b Fs(\))p Fr(:)448 1365 y Fj(Lemma)33 b(4.1)46 b Fs(\(a\))41 b Fo(The)g(family)g Fp(f)p Fr(C)1742 1379 y Fn(0)1782 1365 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\))15 b Fp(j)g Fr(Y)60 b Fp(2)39 b Fm(G)19 b Fs(\()p Fr(X)8 b Fs(\))p Fp(g)47 b Fo(of)40 b Fr(C)2815 1332 y Fq(\003)2854 1365 y Fo(-sub)-5 b(algebr)g(as)42 b(of)448 1478 y Fr(C)513 1493 y Fz(b)548 1478 y Fs(\()p Fr(X)7 b Fs(\))33 b Fo(is)g(line)-5 b(arly)34 b(indep)-5 b(endent.)448 1591 y Fs(\(b\))33 b Fo(If)g Fp(F)h(\032)25 b Fm(G)19 b Fs(\()p Fr(X)8 b Fs(\))39 b Fo(is)32 b(\014nite)h(then)g Fr(C)1756 1558 y Fz(X)1749 1615 y Fn(0)1823 1591 y Fs(\()p Fp(F)9 b Fs(\))34 b Fo(is)e(a)h(close)-5 b(d)34 b(subsp)-5 b(ac)g(e)34 b(of)f Fr(C)2979 1606 y Fz(bu)3054 1591 y Fs(\()p Fr(X)7 b Fs(\))p Fo(.)448 1704 y Fs(\(c\))33 b Fo(F)-7 b(or)33 b(e)-5 b(ach)32 b Fr(Y)5 b(;)15 b(Z)32 b Fp(2)25 b Fm(G)19 b Fs(\()q Fr(X)7 b Fs(\))38 b Fo(the)32 b(set)g Fr(C)1853 1718 y Fn(0)1892 1704 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\))e Fp(\001)f Fr(C)2288 1718 y Fn(0)2328 1704 y Fs(\()p Fr(X=)-5 b(Z)7 b Fs(\))32 b Fo(is)g(a)g(dense)g(sub)-5 b(algebr)g(a)448 1817 y(of)33 b Fr(C)620 1831 y Fn(0)660 1817 y Fs(\()p Fr(X=)p Fs(\()p Fr(Y)41 b Fp(\\)20 b Fr(Z)7 b Fs(\)\))p Fo(.)448 1977 y Fj(Pro)s(of.)48 b Fs(W)-8 b(e)41 b(giv)m(e)g(a)g (detailed)f(pro)s(of)g(of)g(this)g(simple)e(lemma)i(b)s(ecause)h(the)g (same)448 2090 y(argumen)m(t)22 b(will)d(b)s(e)i(used)g(later)h(on)f (in)f(order)h(to)i(pro)m(v)m(e)f(Theorem)f(4.5.)39 b(Let)22 b Fp(F)35 b(\032)25 b Fm(G)19 b Fs(\()p Fr(X)8 b Fs(\))448 2203 y(b)s(e)25 b(\014nite)f(and)h(for)g(eac)m(h)h Fr(Y)45 b Fp(2)25 b(F)35 b Fs(let)25 b Fr(')1768 2217 y Fz(Y)1854 2203 y Fp(2)g Fr(C)2005 2217 y Fn(0)2044 2203 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\).)39 b(Denote)27 b Fr(')2743 2170 y Fz(Y)2829 2203 y Fs(:)f Fr(X=)5 b(Y)46 b Fp(!)25 b Fm(C)49 b Fs(the)448 2316 y(function)23 b(suc)m(h)g(that)i Fr(')1246 2330 y Fz(Y)1332 2316 y Fs(=)g Fr(')1487 2283 y Fz(Y)1555 2316 y Fp(\016)7 b Fr(\031)1659 2330 y Fz(Y)1720 2316 y Fs(.)39 b(Then)22 b(for)i(eac)m(h)h Fr(!)j Fp(2)d Fr(X)31 b Fs(one)24 b(has)f(\()p Fr(\034)3012 2330 y Fz(!)3063 2316 y Fr(')3122 2330 y Fz(Y)3183 2316 y Fs(\)\()p Fr(x)p Fs(\))j(=)448 2429 y Fr(')507 2396 y Fz(Y)569 2429 y Fs(\()p Fr(\031)656 2443 y Fz(Y)716 2429 y Fs(\()p Fr(x)p Fs(\))c(+)f Fr(\031)1004 2443 y Fz(Y)1065 2429 y Fs(\()p Fr(!)s Fs(\)\))32 b(for)g(all)f Fr(x)c Fp(2)h Fr(X)7 b Fs(.)45 b(Hence,)33 b(if)e(w)m(e)h(set)h Fp(F)2577 2443 y Fz(!)2655 2429 y Fs(=)27 b Fp(f)p Fr(Y)48 b Fp(2)28 b(F)c(j)15 b Fr(!)31 b Fp(2)c Fr(Y)20 b Fp(g)p Fs(,)448 2542 y(then)1405 2655 y(lim)1377 2716 y Fz(\025)p Fq(!1)1575 2655 y Fr(\034)1615 2670 y Fz(\025!)1706 2655 y Fs([)1746 2568 y Fl(X)1731 2765 y Fz(Y)c Fq(2F)1907 2655 y Fr(')1966 2669 y Fz(Y)2027 2655 y Fs(])26 b(=)2208 2568 y Fl(X)2174 2765 y Fz(Y)15 b Fq(2F)2327 2773 y Fi(!)2388 2655 y Fr(')2447 2669 y Fz(Y)448 2655 y Fs(\(4.4\))448 2905 y(p)s(oin)m(t)m(wise)29 b(on)i Fr(X)7 b Fs(.)41 b(In)29 b(particular)1466 3016 y Fl(\015)1466 3070 y(\015)1466 3125 y(\015)1466 3179 y(\015)1466 3234 y(\015)1550 3089 y(X)1517 3285 y Fz(Y)15 b Fq(2F)1670 3293 y Fi(!)1730 3175 y Fr(')1789 3189 y Fz(Y)1851 3016 y Fl(\015)1851 3070 y(\015)1851 3125 y(\015)1851 3179 y(\015)1851 3234 y(\015)1926 3175 y Fp(\024)2022 3016 y Fl(\015)2022 3070 y(\015)2022 3125 y(\015)2022 3179 y(\015)2022 3234 y(\015)2088 3089 y(X)2073 3285 y Fz(Y)g Fq(2F)2249 3175 y Fr(')2308 3189 y Fz(Y)2369 3016 y Fl(\015)2369 3070 y(\015)2369 3125 y(\015)2369 3179 y(\015)2369 3234 y(\015)448 3175 y Fs(\(4.5\))448 3464 y(where)30 b Fp(k)q(\001k)h Fs(is)e(the)i(sup)e(norm.)589 3577 y(Let)35 b(us)f(pro)m(v)m(e)g(that)h(there)g(is)e(a)h(n)m(um)m(b)s (er)f Fr(C)41 b Fs(suc)m(h)33 b(that)i(for)f(all)f Fr(Z)38 b Fp(2)31 b(F)44 b Fs(and)33 b(all)448 3690 y Fp(f)p Fr(')552 3704 y Fz(Y)614 3690 y Fp(g)e Fs(as)f(ab)s(o)m(v)m(e)1517 3880 y Fp(k)p Fr(')1621 3894 y Fz(Z)1678 3880 y Fp(k)c(\024)f Fr(C)1931 3721 y Fl(\015)1931 3776 y(\015)1931 3830 y(\015)1931 3885 y(\015)1931 3939 y(\015)1997 3794 y(X)1982 3991 y Fz(Y)16 b Fq(2F)2158 3880 y Fr(')2217 3894 y Fz(Y)2278 3721 y Fl(\015)2278 3776 y(\015)2278 3830 y(\015)2278 3885 y(\015)2278 3939 y(\015)2344 3880 y Fr(:)-1921 b Fs(\(4.6\))448 4131 y(This)29 b(clearly)h(implies)d(\(a\))32 b(and)d(\(b\).)41 b(If)30 b(the)h(set)1211 4343 y Fr(Z)1273 4357 y Fn(0)1338 4343 y Fs(=)25 b Fr(Z)i Fp(n)1620 4256 y Fl([)1588 4454 y Fz(Y)16 b Fq(6)p Fn(=)p Fz(Z)1768 4343 y Fr(Y)45 b Fs(=)1993 4256 y Fl(\\)1962 4454 y Fz(Y)15 b Fq(6)p Fn(=)p Fz(Z)2126 4343 y Fs([)p Fr(Z)27 b Fp(n)20 b Fs(\()p Fr(Y)2429 4256 y Fl(\\)2545 4343 y Fr(Z)7 b Fs(\)])448 4638 y(is)26 b(not)g(empt)m(y)h(then)f(\(4.5\))i(with)d(a)h (c)m(hoice)i Fr(!)g Fp(2)d Fr(Z)2157 4652 y Fn(0)2222 4638 y Fs(giv)m(es)i(\(4.6\))h(\(with)d Fr(C)32 b Fs(=)25 b(1\).)40 b(Since)448 4751 y Fr(Y)34 b Fp(\\)14 b Fr(Z)34 b Fs(are)27 b(linear)f(subspaces)g(of)i Fr(Z)34 b Fs(one)27 b(has)g Fr(Z)2097 4765 y Fn(0)2162 4751 y Fs(=)e Fp(;)i Fs(if)g(and)f(only)h(if)f(there)h(is)g Fr(Y)45 b Fp(2)25 b(F)448 4863 y Fs(suc)m(h)36 b(that)h Fr(Z)42 b Fp(\032)35 b Fr(Y)56 b Fs(strictly)-8 b(.)58 b(This)34 b(cannot)j(happ)s(en)e(if)g Fr(Z)42 b Fs(is)36 b(a)g(maximal)g(elemen)m(t)448 4976 y(in)k Fp(F)9 b Fs(,)45 b(hence)c(\(4.6\))i(holds)d(for)h(suc)m(h)f (elemen)m(ts.)74 b(Let)42 b Fp(F)2493 4990 y Fn(1)2574 4976 y Fs(b)s(e)e(the)i(set)f(of)h Fr(Y)63 b Fp(2)43 b(F)1897 5225 y Fs(14)p eop %%Page: 15 15 15 14 bop 448 573 a Fs(whic)m(h)29 b(are)h(not)f(maximal)g(elemen)m(ts) h(in)e Fp(F)9 b Fs(.)41 b(Then)28 b(w)m(e)i(clearly)f(get)2816 496 y Fl(\015)2816 550 y(\015)2867 505 y(P)2963 600 y Fz(Y)15 b Fq(2F)3116 609 y Fd(1)3170 573 y Fr(')3229 587 y Fz(Y)3291 496 y Fl(\015)3291 550 y(\015)3366 573 y Fp(\024)448 695 y Fr(C)513 709 y Fn(1)568 617 y Fl(\015)568 672 y(\015)618 626 y(P)714 721 y Fz(Y)h Fq(2F)894 695 y Fr(')953 709 y Fz(Y)1015 617 y Fl(\015)1015 672 y(\015)1100 695 y Fs(for)36 b(some)g(constan)m(t)h Fr(C)1915 709 y Fn(1)1954 695 y Fs(.)56 b(By)36 b(what)f(w)m(e)h(already)g(pro)m(v)m (ed)f(w)m(e)h(see)448 807 y(then)30 b(that)h(\(4.6\))i(holds)28 b(for)j(the)f(maximal)f(elemen)m(ts)i Fr(Z)37 b Fs(of)30 b Fp(F)2613 821 y Fn(1)2653 807 y Fs(,)h(etc.)589 920 y(W)-8 b(e)37 b(no)m(w)e(pro)m(v)m(e)i(\(c\).)56 b(Let)36 b Fr(E)j Fs(=)33 b(\()p Fr(X=)5 b(Y)21 b Fs(\))j Fp(\002)f Fs(\()p Fr(X=)-5 b(Z)7 b Fs(\))36 b(equipp)s(ed)d(with)h(the)i(direct) 448 1033 y(sum)30 b(v)m(ector)j(space)e(structure.)42 b(If)31 b Fr(')26 b Fp(2)g Fr(C)1914 1047 y Fn(0)1953 1033 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\))31 b(and)g Fr( )e Fp(2)d Fr(C)2671 1047 y Fn(0)2710 1033 y Fs(\()p Fr(X=)-5 b(Z)7 b Fs(\))32 b(then)e Fr(')21 b Fp(\012)g Fr( )448 1146 y Fs(denotes)46 b(the)f(function)f(\()p Fr(s;)15 b(t)p Fs(\))51 b Fp(7\000)-15 b(!)50 b Fr(')p Fs(\()p Fr(s)p Fs(\))p Fr( )s Fs(\()p Fr(t)p Fs(\),)h(whic)m(h)44 b(b)s(elongs)g(to)i Fr(C)2988 1160 y Fn(0)3027 1146 y Fs(\()p Fr(E)5 b Fs(\).)87 b(The)448 1259 y(subspace)28 b(generated)h(b)m(y)f(the)g(functions)f(of)h(the)h(form)f Fr(')16 b Fp(\012)f Fr( )32 b Fs(is)27 b(dense)h(in)f Fr(C)3131 1273 y Fn(0)3170 1259 y Fs(\()p Fr(E)5 b Fs(\))29 b(b)m(y)448 1372 y(the)e(Stone-W)-8 b(eierstrass)28 b(theorem.)39 b(Let)28 b Fr(F)39 b Fs(b)s(e)26 b(a)h(linear)e(subspace)h(of)h Fr(E)5 b Fs(.)39 b(Since)26 b(eac)m(h)448 1485 y(function)d(in)g Fr(C)963 1499 y Fn(0)1002 1485 y Fs(\()p Fr(F)13 b Fs(\))25 b(extends)f(to)g(a)h(function)d(in)h Fr(C)2181 1499 y Fn(0)2220 1485 y Fs(\()p Fr(E)5 b Fs(\))25 b(w)m(e)g(see)f(that)h(the)f (restrictions)448 1598 y(\()p Fr(')d Fp(\012)f Fr( )s Fs(\))26 b Fp(j)g Fr(F)43 b Fs(generate)32 b(a)f(dense)f(linear)f (subspace)g(of)i Fr(C)2418 1612 y Fn(0)2457 1598 y Fs(\()p Fr(F)13 b Fs(\).)589 1711 y(Let)33 b(us)f(denote)h(b)m(y)f Fr(\031)k Fs(the)d(map)f Fr(x)c Fp(7\000)-15 b(!)28 b Fs(\()p Fr(\031)2089 1725 y Fz(Y)2150 1711 y Fs(\()p Fr(x)p Fs(\))p Fr(;)15 b(\031)2364 1725 y Fz(Z)2422 1711 y Fs(\()p Fr(x)p Fs(\)\),)34 b(so)f Fr(\031)i Fs(is)d(a)h(linear)d(map) 448 1824 y(from)c Fr(X)34 b Fs(to)27 b Fr(E)k Fs(with)25 b(k)m(ernel)h Fr(V)45 b Fs(=)25 b Fr(Y)32 b Fp(\\)12 b Fr(Z)7 b Fs(.)38 b(Let)27 b Fr(F)39 b Fs(b)s(e)26 b(the)g(range)h(of) f Fr(\031)s Fs(.)39 b(Then)26 b(there)g(is)448 1937 y(a)k(linear)e (bijectiv)m(e)i(map)k(~)-50 b Fr(\031)28 b Fs(:)e Fr(X=V)45 b Fp(!)26 b Fr(F)42 b Fs(suc)m(h)29 b(that)i Fr(\031)d Fs(=)i(~)-50 b Fr(\031)22 b Fp(\016)d Fr(\031)2670 1951 y Fz(V)2730 1937 y Fs(.)41 b(So)29 b Fr(\022)f Fp(7\000)-16 b(!)26 b Fr(\022)20 b Fp(\016)k Fs(~)-50 b Fr(\031)33 b Fs(is)448 2049 y(an)27 b(isometric)f(isomorphism)e(of)i Fr(C)1649 2063 y Fn(0)1689 2049 y Fs(\()p Fr(F)13 b Fs(\))27 b(on)m(to)h Fr(C)2123 2063 y Fn(0)2162 2049 y Fs(\()p Fr(X=V)21 b Fs(\).)40 b(Hence)28 b(for)e Fr(')g Fp(2)f Fr(C)3129 2063 y Fn(0)3168 2049 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\))448 2162 y(and)31 b Fr( )e Fp(2)d Fr(C)866 2176 y Fn(0)906 2162 y Fs(\()p Fr(X=)-5 b(Z)7 b Fs(\))31 b(the)g(function)f Fr(\022)e Fs(=)e(\()p Fr(')c Fp(\012)e Fr( )s Fs(\))h Fp(\016)26 b Fs(~)-50 b Fr(\031)34 b Fs(b)s(elongs)c(to)i Fr(C)2854 2176 y Fn(0)2893 2162 y Fs(\()p Fr(X=V)21 b Fs(\),)32 b(it)e(has)448 2275 y(the)d(prop)s(ert)m(y)f Fr(\022)15 b Fp(\016)e Fr(\031)1137 2289 y Fz(V)1223 2275 y Fs(=)25 b Fr(')13 b Fp(\016)g Fr(\031)1501 2289 y Fz(Y)1574 2275 y Fp(\001)g Fr( )j Fp(\016)d Fr(\031)1797 2289 y Fz(Z)1853 2275 y Fs(,)28 b(and)d(the)i(functions)e(of)i(this)e (form)i(generate)448 2388 y(a)k(dense)f(linear)f(subspace)h(of)g Fr(C)1573 2402 y Fn(0)1613 2388 y Fs(\()p Fr(X=V)20 b Fs(\).)p 3371 2380 67 67 v 589 2551 a(W)-8 b(e)33 b(sa)m(y)f(that)f Fp(F)36 b(\032)27 b Fm(G)19 b Fs(\()p Fr(X)8 b Fs(\))37 b(is)30 b Fp(\\)p Fo(-stable)38 b Fs(if)30 b Fr(Y)5 b(;)15 b(Z)34 b Fp(2)26 b(F)36 b(\))26 b Fr(Y)41 b Fp(\\)20 b Fr(Z)33 b Fp(2)27 b(F)40 b Fs(\(so)32 b Fp(F)40 b Fs(is)31 b(a)448 2664 y(generalized)24 b(\015ag)g(of)h(subspaces)e(of)h Fr(X)7 b Fs(\).)40 b(Suc)m(h)23 b(a)i Fp(F)33 b Fs(is)23 b(a)i(semilattice)e(when)h(equipp)s(ed)448 2777 y(with)31 b(the)g(order)h(relation)e(giv)m(en)i(b)m(y)f(inclusion.)42 b(W)-8 b(e)33 b(denote)f(b)m(y)f Fm(F)5 b Fs(\()p Fr(X)j Fs(\))33 b(=)27 b Fm(F)5 b Fs(\()q Fm(G)25 b Fs(\()p Fr(X)7 b Fs(\))q(\))448 2890 y(the)31 b(set)g(of)f(\014nite)g Fp(\\)p Fs(-stable)g(subsets)f(of)i Fm(G)19 b Fs(\()p Fr(X)8 b Fs(\).)448 3077 y Fj(Corollary)35 b(4.2)46 b Fo(If)g Fp(F)58 b(2)49 b Fm(F)5 b Fs(\()p Fr(X)j Fs(\))52 b Fo(then)46 b Fr(C)1976 3044 y Fz(X)1969 3102 y Fn(0)2043 3077 y Fs(\()p Fp(F)9 b Fs(\))46 b Fo(is)g(a)g Fr(C)2508 3044 y Fq(\003)2547 3077 y Fo(-sub)-5 b(algebr)g(a)47 b(of)f Fr(C)3209 3092 y Fz(bu)3284 3077 y Fs(\()p Fr(X)7 b Fs(\))448 3190 y Fo(e)-5 b(quipp)g(e)g(d)42 b(with)f(a)f(natur)-5 b(al)42 b(structur)-5 b(e)40 b(of)g Fp(F)9 b Fo(-gr)-5 b(ade)g(d)42 b Fr(C)2402 3157 y Fq(\003)2441 3190 y Fo(-algebr)-5 b(a.)65 b(This)40 b(algebr)-5 b(a)41 b(is)448 3303 y(unital)34 b(if)e(and)h(only)h(if)e Fr(X)h Fp(2)25 b(F)9 b Fo(.)42 b(F)-7 b(or)33 b Fp(F)1831 3317 y Fn(1)1871 3303 y Fr(;)15 b Fp(F)1976 3317 y Fn(2)2042 3303 y Fp(2)25 b Fm(F)t Fs(\()q Fr(X)7 b Fs(\))39 b Fo(one)33 b(has)1214 3521 y Fr(C)1286 3483 y Fz(X)1279 3543 y Fn(0)1353 3521 y Fs(\()p Fp(F)1453 3535 y Fn(1)1493 3521 y Fs(\))1543 3435 y Fl(\\)1660 3521 y Fr(C)1732 3483 y Fz(X)1725 3543 y Fn(0)1799 3521 y Fs(\()p Fp(F)1899 3535 y Fn(2)1939 3521 y Fs(\))26 b(=)e Fr(C)2167 3483 y Fz(X)2160 3543 y Fn(0)2234 3521 y Fs(\()p Fp(F)2334 3535 y Fn(1)2390 3435 y Fl(\\)2506 3521 y Fp(F)2571 3535 y Fn(2)2611 3521 y Fs(\))p Fr(:)-2223 b Fs(\(4.7\))448 3749 y Fo(In)33 b(p)-5 b(articular,)35 b(one)e(has)g Fr(C)1417 3716 y Fz(X)1410 3773 y Fn(0)1484 3749 y Fs(\()p Fp(F)1584 3763 y Fn(1)1624 3749 y Fs(\))26 b Fp(\032)f Fr(C)1853 3716 y Fz(X)1846 3773 y Fn(0)1920 3749 y Fs(\()p Fp(F)2020 3763 y Fn(2)2060 3749 y Fs(\))33 b Fo(if)f(and)i(only)f(if)f Fp(F)2743 3763 y Fn(1)2808 3749 y Fp(\032)25 b(F)2969 3763 y Fn(2)3009 3749 y Fo(.)448 3990 y Fj(4.2.)38 b Fs(W)-8 b(e)22 b(are)f(no)m(w)g(ready)f(to)h(de\014ne)f(the)h(noncomm)m (utativ)m(e)h(v)m(ersions)e(of)g(the)h(algebras)448 4103 y Fr(C)513 4117 y Fn(0)553 4103 y Fs(\()p Fr(X=)5 b(Y)20 b Fs(\):)53 b(they)36 b(are)h(crossed)f(pro)s(ducts)f(of)h(algebras)g (of)h(the)f(form)g Fr(C)2997 4117 y Fn(0)3036 4103 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\))36 b(b)m(y)448 4216 y(the)41 b(natural)e(action)h Fr(\034)50 b Fs(of)40 b(the)g(additiv)m(e)f(group) h Fr(X)7 b Fs(.)70 b(These)40 b(algebras)g(ha)m(v)m(e)h(b)s(een)448 4329 y(\014rst)e(in)m(tro)s(duced,)i(in)e(a)h(rather)g(di\013eren)m(t)f (form,)j(b)m(y)e(P)m(erry)-8 b(,)43 b(Sigal)c(and)g(Simon)f(in)448 4442 y([PSS].)k(The)f(connection)h(b)s(et)m(w)m(een)g(our)g(form)m (ulation)e(and)h(theirs)g(is)g(clari\014ed)f(b)m(y)448 4554 y(Prop)s(osition)e(4.7)j(b)s(elo)m(w.)69 b(See)40 b(also)g(the)g(in)m(tro)s(ductions)e(of)i(c)m(hapters)g(8)h(and)e(9)h (in)448 4667 y([ABG].)448 4855 y Fj(De\014nition)c(4.3)46 b Fo(If)28 b Fr(Y)49 b Fo(is)29 b(a)g(subsp)-5 b(ac)g(e)30 b(of)f Fr(X)37 b Fo(then)29 b Fv(C)2313 4822 y Fz(X)2380 4855 y Fs(\()p Fr(Y)21 b Fs(\))k(=)g Fr(C)2710 4869 y Fn(0)2750 4855 y Fs(\()p Fr(X=)5 b(Y)20 b Fs(\))12 b Fk(o)g Fr(X)37 b Fo(is)29 b(the)448 4968 y Fr(C)520 4935 y Fq(\003)559 4968 y Fo(-sub)-5 b(algebr)g(a)31 b(of)e Fv(B)p Fs(\()p Fr(X)7 b Fs(\))31 b Fo(obtaine)-5 b(d)31 b(as)f(norm)g(closur)-5 b(e)30 b(of)g Fr(C)2574 4982 y Fn(0)2614 4968 y Fs(\()p Fr(X=)5 b(Y)20 b Fs(\))13 b Fp(\001)g Fr(C)2998 4982 y Fn(0)3039 4968 y Fs(\()p Fr(X)3156 4935 y Fq(\003)3196 4968 y Fs(\))p Fo(.)41 b(F)-7 b(or)1897 5225 y Fs(15)p eop %%Page: 16 16 16 15 bop 448 573 a Fo(e)-5 b(ach)33 b(subset)f Fp(F)j(\032)25 b Fm(G)19 b Fs(\()p Fr(X)8 b Fs(\))38 b Fo(let)32 b Fv(C)1561 540 y Fz(X)1628 573 y Fs(\()p Fp(F)9 b Fs(\))33 b Fo(b)-5 b(e)32 b(the)h(line)-5 b(ar)33 b(subsp)-5 b(ac)g(e)33 b(of)f Fv(B)p Fs(\()p Fr(X)7 b Fs(\))34 b Fo(gener)-5 b(ate)g(d)448 686 y(by)33 b(the)g(algebr)-5 b(as)34 b Fv(C)1124 653 y Fz(X)1191 686 y Fs(\()p Fr(Y)21 b Fs(\))32 b Fo(with)i Fr(Y)45 b Fp(2)25 b(F)9 b Fo(,)33 b(so)1509 896 y Fv(C)1570 858 y Fz(X)1637 896 y Fs(\()p Fp(F)9 b Fs(\))27 b(=)1918 810 y Fl(X)1903 1007 y Fz(Y)16 b Fq(2F)2079 896 y Fv(C)2140 858 y Fz(X)2208 896 y Fs(\()p Fr(Y)k Fs(\))p Fr(:)-1928 b Fs(\(4.8\))589 1225 y(Observ)m(e)33 b(that)g Fv(C)1201 1192 y Fz(X)1269 1225 y Fs(\()p Fp(F)9 b Fs(\))30 b(=)f([)-12 b([)p Fr(C)1653 1192 y Fz(X)1646 1249 y Fn(0)1721 1225 y Fs(\()p Fp(F)9 b Fs(\))23 b Fp(\001)f Fr(C)2000 1239 y Fn(0)2039 1225 y Fs(\()p Fr(X)2156 1192 y Fq(\003)2196 1225 y Fs(\)])-12 b(])31 b(=)e([)-12 b([)p Fr(C)2503 1239 y Fn(0)2543 1225 y Fs(\()p Fr(X)2660 1192 y Fq(\003)2700 1225 y Fs(\))22 b Fp(\001)g Fr(C)2876 1192 y Fz(X)2869 1249 y Fn(0)2943 1225 y Fs(\()p Fp(F)9 b Fs(\)])-12 b(].)50 b(So)32 b(for)448 1338 y Fp(\\)p Fs(-stable)e Fp(F)40 b Fs(w)m(e)31 b(ha)m(v)m(e)g Fv(C)1311 1305 y Fz(X)1379 1338 y Fs(\()p Fp(F)9 b Fs(\))26 b(=)f Fr(C)1717 1305 y Fz(X)1710 1362 y Fn(0)1784 1338 y Fs(\()p Fp(F)9 b Fs(\))21 b Fk(o)f Fr(X)7 b Fs(.)589 1498 y(T)-8 b(o)25 b(eac)m(h)f(v)m(ector)h(subspace)f Fr(Y)43 b Fs(of)24 b Fr(X)31 b Fs(w)m(e)24 b(ha)m(v)m(e)h(th)m(us)e(asso)s(ciated)h(a)g Fr(C)2943 1465 y Fq(\003)2982 1498 y Fs(-subalgebra)448 1611 y Fv(C)509 1578 y Fz(X)577 1611 y Fs(\()p Fr(Y)c Fs(\))31 b(of)f Fv(B)p Fs(\()p Fr(X)7 b Fs(\).)42 b(The)30 b(only)f(one)i(whic)m(h)e(is)h(ab)s(elian)e(is)1102 1807 y Fv(C)1163 1770 y Fz(X)1230 1807 y Fs(\()p Fr(X)7 b Fs(\))27 b(=)e Fr(C)1570 1821 y Fn(0)1609 1807 y Fs(\()p Fr(X)1726 1770 y Fq(\003)1766 1807 y Fs(\))h(=)f Fp(f)p Fr(')p Fs(\()p Fr(P)13 b Fs(\))27 b Fp(j)e Fr(')h Fp(2)f Fr(C)2481 1821 y Fn(0)2520 1807 y Fs(\()p Fr(X)2637 1770 y Fq(\003)2677 1807 y Fs(\))p Fp(g)p Fr(:)-2334 b Fs(\(4.9\))448 2004 y(The)26 b(algebra)h Fv(C)1007 1971 y Fz(X)1074 2004 y Fs(\()p Fr(O)s Fs(\))g(is)f(generated)h(b)m(y)g Fr(C)1929 2018 y Fn(0)1968 2004 y Fs(\()p Fr(X)7 b Fs(\))12 b Fp(\001)g Fr(C)2234 2018 y Fn(0)2276 2004 y Fs(\()p Fr(X)2393 1971 y Fq(\003)2433 2004 y Fs(\))27 b(and,)f(since)g(the)h (op)s(erators)448 2117 y(of)k(the)f(form)g Fr(')p Fs(\()p Fr(Q)p Fs(\))p Fr( )s Fs(\()p Fr(P)13 b Fs(\)\))33 b(with)c Fr(')d Fp(2)f Fr(C)1838 2131 y Fn(0)1877 2117 y Fs(\()p Fr(X)7 b Fs(\))p Fr(;)15 b( )31 b Fp(2)24 b Fr(C)2309 2131 y Fn(0)2349 2117 y Fs(\()p Fr(X)2466 2084 y Fq(\003)2506 2117 y Fs(\))31 b(are)f(compact,)i(w)m(e)f(ha)m(v)m(e)1624 2314 y Fv(C)1685 2276 y Fz(X)1753 2314 y Fs(\()p Fr(O)s Fs(\))25 b(=)g Fv(K)p Fs(\()p Fr(X)7 b Fs(\))p Fr(:)-1812 b Fs(\(4.10\))589 2510 y(The)32 b(algebras)h(whic)m(h)e(pla)m(y)h(the)h (main)e(r^)-45 b(ole)32 b(in)f(the)i Fr(N)10 b Fs(-b)s(o)s(dy)31 b(problem)g(\(as)i(pre-)448 2623 y(sen)m(ted)g(in)e(c)m(h.)46 b(9)33 b(of)f([ABG]\))i(are)e(of)h(the)f(form)g Fv(C)2198 2590 y Fz(X)2265 2623 y Fs(\()p Fp(F)9 b Fs(\))33 b(with)e(\014nite)g Fp(F)42 b Fs(and)31 b(will)f(b)s(e)448 2736 y(sudied)f(in)g(this)g (section.)41 b(The)30 b(next)g(one)h(is)e(dev)m(oted)j(to)f(the)f(case) i Fp(F)i Fs(=)25 b Fm(G)19 b Fs(\()q Fr(X)7 b Fs(\))q(.)589 2849 y(W)-8 b(e)46 b(shall)d(need)h(an)g(extension)g(of)h(the)g (automorphism)e Fr(\034)2714 2863 y Fz(x)2802 2849 y Fs(of)h Fr(C)2984 2864 y Fz(b)3018 2849 y Fs(\()p Fr(X)7 b Fs(\))46 b(to)f(an)448 2962 y(automorphism)29 b(of)h Fv(B)p Fs(\()p Fr(X)7 b Fs(\):)42 b(w)m(e)30 b(set)h Fr(\034)1775 2976 y Fz(x)1819 2962 y Fs([)p Fr(S)5 b Fs(])25 b(=)g(e)2092 2929 y Fz(i)p Fq(h)p Fz(x;P)10 b Fq(i)2289 2962 y Fr(S)5 b Fs(e)2390 2929 y Fq(\000)p Fz(i)p Fq(h)p Fz(x;P)10 b Fq(i)2672 2962 y Fs(for)30 b(eac)m(h)h Fr(x)25 b Fp(2)g Fr(X)38 b Fs(and)448 3075 y Fr(S)31 b Fp(2)24 b Fv(B)p Fs(\()p Fr(X)7 b Fs(\).)42 b(Observ)m(e)31 b(that)g(for)f Fr(')25 b Fp(2)g Fr(C)1848 3089 y Fn(0)1888 3075 y Fs(\()p Fr(X=)5 b(Y)20 b Fs(\))31 b(and)f Fr( )f Fp(2)24 b Fr(C)2602 3089 y Fn(0)2642 3075 y Fs(\()p Fr(X)2759 3042 y Fq(\003)2799 3075 y Fs(\))31 b(one)f(has)1162 3271 y Fr(\034)1202 3285 y Fz(x)1246 3271 y Fs([)p Fr(')p Fs(\()p Fr(Q)1437 3285 y Fz(Y)1499 3271 y Fs(\))p Fr( )s Fs(\()p Fr(P)13 b Fs(\)])27 b(=)e Fr(')p Fs(\()p Fr(Q)2051 3285 y Fz(Y)2132 3271 y Fs(+)20 b Fr(\031)2275 3285 y Fz(Y)2336 3271 y Fs(\()p Fr(x)p Fs(\)\))p Fr( )s Fs(\()p Fr(P)13 b Fs(\))p Fr(:)448 3468 y Fs(This)29 b(immediately)g(giv)m(es)h(the)h(next)f(lemma.)448 3649 y Fj(Lemma)j(4.4)46 b Fo(\(i\))33 b(If)f Fr(y)c Fp(2)d Fr(Y)53 b Fo(and)33 b Fr(S)e Fp(2)24 b Fv(C)1902 3616 y Fz(X)1970 3649 y Fs(\()p Fr(Y)c Fs(\))33 b Fo(then)g Fr(\034)2388 3663 y Fz(y)2429 3649 y Fs([)p Fr(S)5 b Fs(])26 b(=)f 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Fr(w)2920 4830 y Fz([)2978 4863 y Fs(:)d Fr(X=)5 b(Y)46 b Fp(!)26 b Fm(C)448 4976 y Fs(b)m(y)31 b Fr(w)642 4943 y Fz([)699 4976 y Fs(=)25 b Fr(w)e Fp(\016)d Fr(p)994 4943 y Fz([)994 5003 y(Z)1051 4976 y Fs(,)30 b(then)g Fr(w)1380 4943 y Fz([)1438 4976 y Fp(2)25 b Fr(C)1589 4990 y Fn(0)1628 4976 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\))30 b(if)g Fr(w)d Fp(2)e Fr(C)2254 4990 y Fn(0)2294 4976 y Fs(\()p Fr(Z)7 b Fs(\))p Fr(:)1897 5225 y Fs(17)p eop %%Page: 18 18 18 17 bop 448 573 a Fj(Prop)s(osition)37 b(4.7)46 b Fo(Ther)-5 b(e)24 b(is)f(a)h(line)-5 b(ar)24 b(c)-5 b(ontinuous)24 b(map)h Fr(C)2554 587 y Fn(0)2593 573 y Fs(\()p Fr(Y)2702 540 y Fq(\003)2741 573 y Fs(\))p Fp(\012)p Fv(K)p Fs(\()p Fr(Z)7 b Fs(\))25 b Fp(!)g Fv(C)3255 540 y Fz(X)3323 573 y Fs(\()p Fr(Y)20 b Fs(\))448 686 y Fo(such)43 b(that)i(for)e(e)-5 b(ach)44 b Fr(u)g Fp(2)f Fr(C)1497 700 y Fn(0)1537 686 y Fs(\()p Fr(Y)1645 653 y Fq(\003)1685 686 y Fs(\))p Fr(;)15 b(v)48 b Fp(2)43 b Fr(C)2021 700 y Fn(0)2061 686 y Fs(\()p Fr(Z)2165 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Fv(C)1083 4659 y Fz(X)1182 4692 y Fo(b)-5 b(e)33 b(de\014ne)-5 b(d)34 b(by)1517 4881 y Fp(P)7 b Fs([)p Fr(T)13 b Fs(])26 b(=)1888 4795 y Fl(M)1825 4996 y Fz(Y)16 b Fq(2)p Fe(P)m Fn(\()p Fz(X)5 b Fn(\))2103 4881 y Fp(P)2166 4895 y Fz(Y)2227 4881 y Fs([)p Fr(T)13 b Fs(])p Fr(:)-1920 b Fs(\(5.7\))1897 5225 y(19)p eop %%Page: 20 20 20 19 bop 448 573 a Fo(Then)33 b Fp(P)41 b Fo(is)32 b(a)h(morphism)i (and)f(its)e(kernel)h(is)g(e)-5 b(qual)33 b(to)g Fv(K)p Fs(\()p Fr(X)7 b Fs(\))p Fo(.)589 749 y Fs(So)36 b Fp(P)42 b Fs(induces)34 b(an)h(em)m(b)s(edding)e(of)j(the)f Fr(C)2092 716 y Fq(\003)2131 749 y Fs(-algebra)h Fv(C)2546 716 y Fz(X)2613 749 y Fr(=)p Fv(K)p Fs(\()p Fr(X)7 b Fs(\))36 b(in)m(to)3108 719 y Fl(e)3102 749 y Fv(C)3163 716 y Fz(X)3231 749 y Fs(.)55 b(W)-8 b(e)448 876 y(shall)29 b(iden)m(tify)g Fv(C)1053 843 y Fz(X)1120 876 y Fr(=)p Fv(K)p Fs(\()p Fr(X)7 b Fs(\))32 b(with)d(a)h(subalgebra)g(of)2263 846 y Fl(e)2257 876 y Fv(C)2318 843 y Fz(X)2386 876 y Fs(:)1608 1093 y Fv(C)1669 1056 y Fz(X)1737 1093 y Fr(=)p Fv(K)p Fs(\()p Fr(X)7 b Fs(\))26 b Fp(\032)2129 1064 y Fl(e)2123 1093 y Fv(C)2184 1056 y Fz(X)2252 1093 y Fr(:)-1829 b Fs(\(5.8\))448 1298 y Fj(5.2.)69 b Fs(W)-8 b(e)41 b(shall)d(mak)m(e)j(here)f(some)g(\014nal)e(remarks)i (concerning)f(the)h(algebra)f Fv(C)3344 1265 y Fz(X)3412 1298 y Fs(.)448 1410 y(First)24 b(w)m(e)h(giv)m(e)g(another)g (description)d(of)j(the)g(maps)f Fp(P)2304 1424 y Fz(Y)2365 1410 y Fs(.)38 b(Observ)m(e)25 b(that)g(b)m(y)f(Theorem)448 1523 y(4.5\(a\))37 b(eac)m(h)f Fr(T)46 b Fp(2)32 b Fv(C)1177 1490 y Fz(X)1244 1523 y Fs(\()p Fm(G)20 b Fs(\()p Fr(X)7 b Fs(\))q(\))41 b(can)35 b(b)s(e)f(written)g(in)f(a)i(unique)e(w)m(a)m (y)j(as)f(a)g(sum)f Fr(T)45 b Fs(=)448 1568 y Fl(P)544 1663 y Fz(Z)5 b Fq(2F)720 1636 y Fr(T)13 b Fs(\()p Fr(Z)7 b Fs(\))31 b(with)e Fp(F)34 b(\032)25 b Fm(G)19 b Fs(\()q Fr(X)7 b Fs(\))37 b(\014nite)29 b(and)h Fr(T)13 b Fs(\()p Fr(Z)7 b Fs(\))25 b Fp(2)g Fv(C)2399 1603 y Fz(X)2467 1636 y Fs(\()p Fr(Z)7 b Fs(\),)31 b 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Fs(\))p Fr(;)448 2742 y Fs(b)m(y)31 b(Lemma)f(4.4.)42 b(In)30 b(other)g(terms,)h(for)f Fr(T)43 b Fs(as)31 b(ab)s(o)m(v)m(e)g (w)m(e)g(ha)m(v)m(e)1450 2947 y Fp(P)1513 2961 y Fz(Y)1574 2947 y Fs([)p Fr(T)13 b Fs(])25 b(=)g Fr(s)20 b Fp(\000)48 b Fs(lim)1965 3008 y Fz(\025)p Fq(!1)2163 2947 y Fr(\034)2203 2962 y Fz(\025!)2294 2947 y Fs([)p Fr(T)13 b Fs(])p Fr(:)-1987 b Fs(\(5.10\))448 3180 y(In)30 b(particular)f(w)m(e)h(get:)448 3367 y Fj(Lemma)j(5.3)46 b Fo(If)29 b Fr(Y)50 b Fo(is)29 b(a)h(one)g(dimensional)h(subsp)-5 b(ac)g(e)31 b(of)e Fr(X)37 b Fo(and)31 b Fr(!)d Fp(2)d Fr(Y)33 b Fp(n)13 b(f)p Fs(0)p Fp(g)32 b Fo(then)448 3480 y(one)h(has)h(for)f(al)5 b(l)33 b Fr(T)38 b Fp(2)25 b Fv(C)1296 3447 y Fz(X)1363 3480 y Fo(:)1450 3684 y Fp(P)1513 3698 y Fz(Y)1574 3684 y Fs([)p Fr(T)13 b Fs(])25 b(=)g Fr(s)20 b Fp(\000)48 b Fs(lim)1965 3746 y Fz(\025)p Fq(!1)2163 3684 y Fr(\034)2203 3699 y Fz(\025!)2294 3684 y Fs([)p Fr(T)13 b Fs(])p Fr(:)-1987 b Fs(\(5.11\))589 4002 y(In)25 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Fz(X)1777 4503 y Fr(C)1842 4517 y Fn(0)1881 4503 y Fs(\()p Fr(X=)5 b(Y)21 b Fs(\),)49 b(and)c(ev)m(erything)g(w)m (e)g(ha)m(v)m(e)h(done)448 4616 y(applies)34 b(to)j Fr(C)946 4583 y Fz(X)939 4640 y Fn(0)1048 4616 y Fs(to)s(o.)57 b(In)35 b(particular,)h(for)f Fr(f)44 b Fp(2)34 b Fr(C)2229 4583 y Fz(X)2222 4640 y Fn(0)2331 4616 y Fs(w)m(e)i(ha)m(v)m(e:)53 b Fr(f)43 b Fp(2)34 b Fr(C)2974 4630 y Fn(0)3013 4616 y Fs(\()p Fr(X)7 b Fs(\))37 b(if)e(and)448 4729 y(only)24 b(if)f(lim)845 4744 y Fz(\025)p Fq(!1)1047 4729 y Fr(f)10 b Fs(\()p Fr(x)e Fs(+)g Fr(\025!)s Fs(\))25 b(=)g(0)f(for)g(eac)m(h)i Fr(!)i Fp(2)d Fr(X)15 b Fp(n)8 b(f)p Fs(0)p Fp(g)p Fs(.)41 b(A)24 b(geometric)h(pro)s(of)f(of)g(this)448 4842 y(not)35 b(ob)m(vious)e(fact)i(\(if)f(dim)14 b Fr(X)39 b(>)31 b Fs(2\))k(has)f(b)s(een)f(sho)m(wn)h(to)h(us)e(b)m(y)h(Radu-Alexandru) 448 4955 y(T)-8 b(o)s(dor.)41 b(W)-8 b(e)31 b(thank)f(him)f(for)h (that.)1897 5225 y(20)p eop %%Page: 21 21 21 20 bop 589 573 a Fs(Certain)24 b(partitions)f(of)h(unit)m(y)g(in)m (tro)s(duced)e(b)m(y)i(F)-8 b(ro)s(ese)26 b(and)d(Herbst)i(in)e([FH])i (ha)m(v)m(e)448 686 y(pro)m(v)m(ed)g(to)f(b)s(e)f(v)m(ery)i(useful)d (in)g(the)i(usual)f(treatmen)m(t)i(of)f Fr(N)10 b Fs(-b)s(o)s(dy)23 b(hamiltonians.)36 b(W)-8 b(e)448 799 y(shall)22 b(brie\015y)f(presen)m (t)i(them)g(and)f(their)g(relation)g(with)g(the)h(algebras)g Fv(C)2923 766 y Fz(X)2990 799 y Fs(\()p Fp(F)9 b Fs(\).)39 b(Belo)m(w)448 912 y(w)m(e)31 b(assume)f(that)h(an)g(euclidean)e(norm)g (is)h(giv)m(en)g(on)g Fr(X)7 b Fs(.)589 1069 y(Let)756 1052 y Fr(\037)843 1069 y Fs(:)31 b Fr(X)39 b Fp(!)30 b Fm(R)43 b Fs(b)s(e)33 b(a)h Fr(C)1512 1036 y Fq(1)1620 1069 y Fs(function,)g(homogeneous)g(of)g(degree)g(zero)h(outside)448 1182 y(the)45 b(unit)e(sphere.)82 b(Since)43 b(the)i(algebra)f Fv(C)2001 1149 y Fz(X)2113 1182 y Fs(is)g(generated)h(b)m(y)f (functions)f(of)i(the)448 1295 y(form)39 b Fr(')p Fs(\()p Fr(Q)p Fs(\))p Fr( )s Fs(\()p Fr(P)13 b Fs(\))41 b(\(or)e Fr( )s Fs(\()p Fr(P)13 b Fs(\))p Fr(')p Fs(\()p Fr(Q)p Fs(\)\))41 b(with)d Fr( )k Fp(2)d(S)7 b Fs(\()p Fr(X)2348 1262 y Fq(\003)2388 1295 y Fs(\),)42 b(it)c(is)g(easy)h(to)h(pro)m(v)m (e)f(that)448 1408 y([)473 1390 y Fr(\037)530 1408 y Fs(\()p Fr(Q)p Fs(\))p Fr(;)15 b(T)e Fs(])27 b Fp(2)e Fv(K)p Fs(\()p Fr(X)7 b Fs(\))31 b(for)f(all)f Fr(T)38 b Fp(2)25 b Fv(C)1669 1375 y Fz(X)1736 1408 y Fs(.)589 1566 y(No)m(w)30 b(let)f Fr(Z)j Fp(\032)25 b Fr(X)37 b Fs(b)s(e)28 b(a)h(subspace)g(and)f(assume)h(that)2486 1548 y Fr(\037)2543 1566 y Fs(\()p Fr(z)t Fs(\))d(=)f(0)30 b(if)e Fr(z)h Fp(2)c Fr(Z)q(;)31 b Fp(j)15 b Fr(z)20 b Fp(j)25 b(\025)448 1678 y Fs(1.)62 b(Then)36 b(for)h(eac)m(h)h Fr(')f Fp(2)g Fr(C)1440 1692 y Fn(0)1479 1678 y Fs(\()p Fr(X=)-5 b(Z)7 b Fs(\))38 b(the)f(function)2297 1661 y Fr(\037)2379 1678 y Fp(\001)25 b Fr(')g Fp(\016)g Fr(\031)2635 1692 y Fz(Z)2729 1678 y Fs(b)s(elongs)36 b(to)i Fr(C)3245 1692 y Fn(0)3284 1678 y Fs(\()p Fr(X)7 b Fs(\))448 1791 y(\(indeed,)29 b(if)e Fr(x)f Fp(!)f(1)j Fs(and)h Fr(\031)1418 1805 y Fz(Z)1474 1791 y Fs(\()p Fr(x)p Fs(\))h(is)d(b)s(ounded,)h(then) g Fr(x=)15 b Fp(j)h Fr(x)f Fp(j)29 b Fs(approac)m(hes)h Fr(Z)7 b Fs(\),)29 b(hence)448 1887 y Fr(\037)505 1904 y Fs(\()p Fr(Q)p Fs(\))p Fr(')p Fs(\()p Fr(Q)813 1918 y Fz(Z)871 1904 y Fs(\))p Fr( )s Fs(\()p Fr(P)13 b Fs(\))35 b(is)d(a)h(compact)h(op)s(erator.)49 b(It)34 b(follo)m(ws)e(that)2694 1887 y Fr(\037)2751 1904 y Fs(\()p Fr(Q)p Fs(\))p Fr(T)46 b Fs(and)33 b Fr(T)3238 1887 y(\037)3295 1904 y Fs(\()p Fr(Q)p Fs(\))448 2017 y(are)e(compact)h(op)s(erators)e(if)g Fr(T)38 b Fp(2)24 b Fv(C)1687 1984 y Fz(X)1755 2017 y Fs(\()p Fr(Z)7 b Fs(\).)589 2175 y(Let)27 b(us)f(\014x)g Fp(F)34 b(2)25 b Fm(F)5 b Fs(\()p Fr(X)j Fs(\))32 b(with)25 b Fr(O)s(;)15 b(X)33 b Fp(2)25 b(F)36 b Fs(and)26 b(let)g Fr(Y)45 b Fp(2)25 b(F)9 b Fr(;)31 b(Y)45 b Fp(6)p Fs(=)25 b Fr(X)7 b Fs(.)40 b(A)26 b Fr(C)3137 2142 y Fq(1)3238 2175 y Fs(func-)448 2288 y(tion)636 2270 y Fr(\037)693 2302 y Fz(Y)782 2288 y Fs(:)h Fr(X)35 b 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Fs(\()p Fr(Q)p Fs(\))h(b)s(elong)e(to)i Fv(K)p Fs(\()p Fr(X)7 b Fs(\))21 b(if)e Fr(T)38 b Fp(2)25 b Fv(C)p Fs(\()p Fp(F)2490 2819 y Fq(0)2481 2879 y Fz(Y)2543 2852 y Fs(\))g(=)2699 2784 y Fl(P)2795 2879 y Fz(Z)5 b Fq(2F)i Fz(;Y)15 b Fq(6\032)p Fz(Z)3154 2852 y Fv(C)3215 2819 y Fz(X)3283 2852 y Fs(\()p Fr(Z)7 b Fs(\).)589 3010 y(Let)37 b Fp(N)49 b Fs(b)s(e)35 b(the)i(set)f(of)g(atoms)h(of)f Fp(F)9 b Fs(.)58 b(An)36 b Fp(F)9 b Fs(-reducing)35 b(partition)g(of)h(unit)m(y)f(on)448 3123 y Fr(X)40 b Fs(is)31 b(a)i(family)e Fp(f)1059 3105 y Fr(\037)1116 3137 y Fz(Y)1177 3123 y Fp(g)1222 3137 y Fz(Y)16 b Fq(2N)1430 3123 y Fs(suc)m(h)32 b(that)1836 3105 y Fr(\037)1893 3137 y Fz(Y)1986 3123 y Fs(is)f Fr(Y)20 b Fs(-reducing)31 b(and)2732 3055 y Fl(P)2828 3150 y Fz(Y)15 b Fq(2N)3018 3105 y Fr(\037)3075 3090 y Fn(2)3075 3150 y Fz(Y)3164 3123 y Fs(=)28 b(1)33 b(on)448 3236 y Fr(X)7 b Fs(.)49 b(In)32 b([FH])i(suc)m(h)f(families)d(are)k (constructed.)48 b(F)-8 b(rom)34 b(\(i\))f(and)f(\(ii\))g(ab)s(o)m(v)m (e)i(w)m(e)f(then)448 3349 y(get:)42 b(if)28 b Fr(S)i Fp(2)25 b Fv(C)p Fs(\()p Fp(F)9 b Fs(\))30 b(and)f(if)f(w)m(e)i(denote) f Fr(S)1862 3363 y Fz(Y)1948 3349 y Fs(=)c Fp(P)2107 3363 y Fz(Y)2168 3349 y Fs([)p Fr(S)5 b Fs(])30 b(its)e(canonical)h (pro)5 b(jection)29 b(on)m(to)448 3462 y Fv(C)p Fs(\()p Fp(F)609 3476 y Fz(Y)671 3462 y Fs(\),)i(then)f(there)g(is)g Fr(K)i Fp(2)25 b Fv(K)p Fs(\()p Fr(X)7 b Fs(\))31 b(suc)m(h)f(that)1315 3664 y Fr(S)g Fs(=)25 b Fr(K)i Fs(+)1712 3578 y Fl(X)1692 3775 y Fz(Y)15 b Fq(2N)1878 3647 y Fr(\037)1935 3678 y Fz(Y)1996 3664 y Fs(\()p Fr(Q)p Fs(\))g Fr(S)2209 3678 y Fz(Y)2285 3647 y Fr(\037)2342 3678 y Fz(Y)2403 3664 y Fs(\()p Fr(Q)p Fs(\))p Fr(:)-2122 b Fs(\(5.12\))589 3935 y(It)39 b(is)f(clear)h(that)h Fp(F)48 b Fs(cannot)39 b(b)s(e)f(replaced)h(b)m(y)f(an)h(in\014nite)e(semilattice)h(in)g(the) 448 4035 y(preceding)f(construction.)63 b(Ho)m(w)m(ev)m(er,)42 b(these)c(partitions)f(can)h(b)s(e)f(used)g(to)i(giv)m(e)f(an)448 4135 y(alternate)32 b(and)d(more)i(elemen)m(tary)g(pro)s(of)f(of)h(the) f(main)g(assertion)g(of)g(Theorem)h(5.2,)448 4234 y(namely)j(that)g Fr(T)47 b Fs(is)33 b(compact)i(if)e Fr(T)44 b Fp(2)31 b Fv(C)1867 4201 y Fz(X)1969 4234 y Fs(and)i(w)24 b Fp(\000)e Fs(lim)2458 4249 y Fz(\025)p Fq(!1)2660 4234 y Fr(\034)2700 4249 y Fz(\025!)2791 4234 y Fs([)p Fr(T)13 b Fs(])32 b(=)f(0)j(for)g(eac)m(h)448 4334 y Fr(!)47 b Fp(2)c Fr(X)35 b Fp(n)28 b(f)p Fs(0)p Fp(g)p Fs(.)75 b(Indeed,)44 b(for)d(eac)m(h)h Fr(")j(>)e Fs(0)f(w)m(e)g(can)f(\014nd)f Fp(F)51 b Fs(as)41 b(ab)s(o)m(v)m(e)i(and)e Fr(S)49 b Fp(2)448 4434 y Fv(C)p Fs(\()p Fp(F)9 b Fs(\))45 b(suc)m(h)f(that)h Fp(k)p Fr(T)33 b Fp(\000)20 b Fr(S)5 b Fp(k)49 b(\024)e Fr(")p Fs(.)83 b(Note)45 b(that)g(w)m(e)f(can)g(assume)g Fr(T)57 b Fs(and)44 b Fr(S)49 b Fs(self-)448 4543 y(adjoin)m(t.)67 b(W)-8 b(rite)40 b Fr(S)45 b Fs(=)1292 4475 y Fl(P)1388 4543 y Fp(f)p Fr(S)5 b Fs(\()p Fr(Z)i Fs(\))15 b Fp(j)g Fr(Z)48 b Fp(2)39 b(F)9 b(g)40 b Fs(with)e Fr(S)5 b Fs(\()p Fr(Z)i Fs(\))41 b Fp(2)e Fv(C)2675 4510 y Fz(X)2743 4543 y Fs(\()p Fr(Z)7 b Fs(\),)41 b(hence)f Fr(S)3266 4557 y Fz(Y)3366 4543 y Fs(=)448 4574 y Fl(P)544 4643 y Fp(f)p Fr(S)5 b Fs(\()p Fr(Z)i Fs(\))15 b Fp(j)g Fr(Z)38 b Fp(2)30 b(F)9 b Fr(;)15 b(Z)37 b Fp(\033)30 b Fr(Y)20 b Fp(g)p Fs(,)35 b(and)e(let)g Fp(F)1906 4657 y Fz(!)1987 4643 y Fs(=)d Fp(f)p Fr(Z)37 b Fp(2)30 b(F)24 b(j)15 b Fr(!)33 b Fp(2)d Fr(Z)7 b Fp(g)33 b Fs(for)h Fr(!)f Fp(2)c Fr(X)h Fp(n)22 b(f)p Fs(0)p Fp(g)p Fs(.)448 4742 y(Then)31 b(s)21 b Fp(\000)g Fs(lim)962 4757 y Fz(\025)p Fq(!1)1164 4742 y Fr(\034)1204 4757 y Fz(\025!)1295 4742 y Fs([)p Fr(S)5 b Fs(])28 b(=)1532 4674 y Fl(P)1628 4769 y Fz(Z)5 b Fq(2F)1778 4777 y Fi(!)1842 4742 y Fr(S)g Fs(\()p Fr(Z)i Fs(\))32 b(and)f(from)g(the)h(F)-8 b(atou)33 b(lemma)f(w)m(e)g(get)448 4786 y Fl(\015)448 4840 y(\015)499 4795 y(P)595 4890 y Fz(Z)5 b Fq(2F)745 4898 y Fi(!)809 4863 y Fr(S)g Fs(\()p Fr(Z)i Fs(\))1009 4786 y Fl(\015)1009 4840 y(\015)1096 4863 y Fp(\024)36 b Fr(":)i Fs(F)-8 b(or)38 b(eac)m(h)g Fr(Y)57 b Fp(2)36 b(N)50 b Fs(w)m(e)38 b(can)f(\014nd)f Fr(!)j Fp(2)d Fr(Y)45 b Fp(n)25 b(f)p Fs(0)p Fp(g)39 b Fs(suc)m(h)d(that)448 4976 y Fr(!)k(=)-55 b Fp(2)26 b Fr(Y)694 4943 y Fq(0)749 4976 y Fs(if)k Fr(Y)906 4943 y Fq(0)955 4976 y Fp(2)c(N)34 b(n)21 b(f)p Fr(Y)g Fp(g)p Fs(.)43 b(Hence)32 b(w)m(e)f(get)h Fp(k)q Fr(S)2108 4990 y Fz(Y)2168 4976 y Fp(k)27 b(\024)f Fr(")p Fs(,)32 b(or)f Fp(\000)p Fr(")26 b Fp(\024)g Fr(S)2840 4990 y Fz(Y)2927 4976 y Fp(\024)g Fr(")p Fs(,)32 b(for)f(eac)m(h)1897 5225 y(21)p eop %%Page: 22 22 22 21 bop 448 573 a Fr(Y)55 b Fp(2)35 b(N)13 b Fs(.)58 b(Then)36 b Fp(\000)p Fr(")f Fp(\024)1321 505 y Fl(P)1432 555 y Fr(\037)1489 587 y Fz(Y)1549 573 y Fs(\()p Fr(Q)p Fs(\))15 b Fr(S)1762 587 y Fz(Y)1839 555 y Fr(\037)1896 587 y Fz(Y)1956 573 y Fs(\()p Fr(Q)p Fs(\))36 b Fp(\024)f Fr(")h Fs(b)s(ecause)2661 505 y Fl(P)2772 555 y Fr(\037)2829 587 y Fz(Y)2889 573 y Fs(\()p Fr(Q)p Fs(\))3031 540 y Fn(2)3107 573 y Fs(=)e(1.)59 b(So)448 672 y(from)36 b(\(5.12\))i(w)m(e) e(see)h(that)f(there)h(is)e Fr(K)41 b Fp(2)34 b Fv(K)p Fs(\()p Fr(X)7 b Fs(\))37 b(suc)m(h)e(that)i Fp(k)p Fr(S)25 b Fp(\000)20 b Fr(K)7 b Fp(k)35 b Fr(<)f(")p Fs(.)58 b(This)448 772 y(implies)28 b Fp(k)p Fr(T)33 b Fp(\000)20 b Fr(K)7 b Fp(k)26 b(\024)f Fs(2)p Fr(")p Fs(,)31 b(whic)m(h)e(pro)m(v) m(es)i(the)g(assertion.)448 1047 y Ft(References)448 1228 y FB([ABG])46 b(W.)38 b(Amrein,)i(A.)e(Boutet)f(de)h(Mon)n(v)n (el,)g(and)g(V.)f(Georgescu,)i Fb(C)2846 1240 y Fa(0)2883 1228 y Fb(-Gr)l(oups,)j(Com-)623 1328 y(mutator)35 b(Metho)l(ds)j(and)e (Sp)l(e)l(ctr)l(al)g(The)l(ory)i(of)f Fx(N)9 b Fb(-Bo)l(dy)37 b(Hamiltonians)p FB(,)g(Birkh\177)-42 b(auser,)623 1428 y(Progress)25 b(in)i(Math.)h(Ser.,)g Fy(135)p FB(,)e(1996.)448 1577 y([DG])46 b(M.)34 b(Damak)f(and)g(V.)g(Georgescu,)g Fx(C)1872 1547 y Fw(\003)1911 1577 y Fb(-A)n(lgebr)l(as)i(R)l(elate)l (d)g(to)f(the)h Fx(N)9 b Fb(-Bo)l(dy)36 b(Pr)l(oblem)623 1677 y(and)30 b(Self-A)l(djoint)h(Op)l(er)l(ators)f(A\016liate)l(d)g (to)g(Them)p FB(,)f(to)e(b)r(e)h(submitted.)448 1826 y([FH])46 b(R.)22 b(G.)h(F)-7 b(ro)r(ese)21 b(and)g(I.)h(Herbst,)h Fb(A)i(New)f(Pr)l(o)l(of)i(of)f(the)g(Mourr)l(e)g(Estimate)p FB(,)e(Duk)n(e)f(Math.)623 1926 y(Journal)k(49,)h(1075-1085)c (\(1982\).)448 2075 y([GI])46 b(V.)26 b(Georgescu)d(and)i(A.)h(Iftimo)n (vici,)f Fx(C)1870 2045 y Fw(\003)1909 2075 y Fb(-A)n(lgebr)l(as)j(of)g (Ener)l(gy)g(Observables)p FB(,)f(in)e(prepa-)623 2175 y(ration.)448 2324 y([PSS])46 b(P)-7 b(.)26 b(P)n(erry)-7 b(,)25 b(I.M.)i(Sigal)e(and)i(B.)f(Simon,)h Fb(Sp)l(e)l(ctr)l(al)i(A)n (nalysis)g(of)h Fx(N)9 b Fb(-Bo)l(dy)29 b(Schr\177)-42 b(odinger)623 2424 y(Op)l(er)l(ators)p FB(,)28 b(Ann.)g(Math.)g(114)e (\(1981\),)g(519-567.)1897 5225 y Fs(22)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9912210646991--