Content-Type: multipart/mixed; boundary="-------------9907080444296" This is a multi-part message in MIME format. ---------------9907080444296 Content-Type: text/plain; name="99-262.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-262.keywords" twisted duality, pairs of projections, CAR-algebra, modular theory ---------------9907080444296 Content-Type: application/postscript; name="preprint401.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="preprint401.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.58 (gTeX distribution 2.2) Copyright 1986, 1994 Radical Eye Software %%Title: carbaum.dvi %%CreationDate: Sat Jun 26 14:20:11 1999 %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: c:/gtex/dvips.exe -M carbaum.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.06.26:0940 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N 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Fv(^)g Fw(p)868 1835 y Ft(1)898 1828 y Fv(^)g Fw(:)e(:)g(:)h Fv(^)h Fw(p)1065 1835 y Fp(n)1096 1828 y Fw(;)-12 1928 y Fy(where)18 b(\012)g(is)g(the)h(F)l(o) q(c)o(k)e(v)m(acuum)i(in)g(the)f(subspace)h(corresp)q(onding)g(to)e Fw(n)g Fy(=)h(0)g(in)h(the)f(de\014nition)h(\(3\))e(and)-12 1985 y Fw(p;)8 b(p)55 1992 y Ft(1)74 1985 y Fw(;)g(:)g(:)g(:)t(;)g(p) 198 1992 y Fp(n)239 1985 y Fv(2)17 b Fw(P)6 b Fx(h)p Fy(.)391 1983 y FB(b)387 1985 y Fw(p)410 1992 y Fp(r)447 1985 y Fy(means)19 b(that)e(the)h(v)o(ector)g Fw(p)935 1992 y Fp(r)972 1985 y Fy(is)h(omitted)f(in)h(the)f(w)o(edge)g(pro)q (duct.)29 b(Finally)l(,)21 b(the)-12 2041 y(F)l(o)q(c)o(k)15 b(represen)o(tation)g Fw(\031)i Fy(is)f(de\014ned)g(b)o(y)561 2141 y Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(f)5 b Fy(\)\))11 b(:=)i Fw(c)p Fy(\()p Fw(P)6 b Fy(\000)p Fw(f)f Fy(\))930 2123 y Fu(\003)960 2141 y Fy(+)10 b Fw(c)p Fy(\()p Fw(P)c(f)f Fy(\))j Fw(;)52 b(f)18 b Fv(2)13 b Fx(h)8 b Fw(:)59 2242 y Fy(In)15 b(the)g(rest)g(of)f(the)h(pap)q(er)h(w)o(e)f(assume)g(that)f (a)h(basis)g(pro)s(jection)g Fw(P)21 b Fy(is)16 b(giv)o(en)f(and)g (when)h(no)f(confusion)-12 2298 y(arises)f(w)o(e)g(will)i(also)e(write) g(simply)h Fw(a)p Fy(\()p Fw(f)5 b Fy(\))14 b(instead)h(of)e Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(f)5 b Fy(\)\).)18 b(T)l(o)c(pro)o(v)o(e)f(t)o(wisted)h(dualit)o(y)h(in)g(Section)g(5)f(w) o(e)-12 2355 y(will)19 b(need)g(an)e(explicit)j(form)o(ula)d(for)g(the) g(v)o(ector)g Fw(a)p Fy(\()p Fw(f)937 2362 y Fp(n)961 2355 y Fy(\))11 b Fv(\001)g Fw(:)d(:)g(:)i Fv(\001)h Fw(a)p Fy(\()p Fw(f)1167 2362 y Ft(1)1187 2355 y Fy(\))d(\012.)26 b(Let)18 b Fw(n;)8 b(k)q(;)g(p)16 b Fy(b)q(e)i(natural)f(n)o(um)o(b)q (ers)-12 2411 y(with)e(2)p Fw(p)10 b Fy(+)h Fw(k)i Fy(=)g Fw(n)j Fy(and)f(de\014ne)h(the)g(follo)o(wing)g(subset)f(of)g(the)g (symmetric)g(group)g Fx(S)1468 2418 y Fp(n)1492 2411 y Fy(:)110 2542 y Fx(S)148 2549 y Fp(n;)6 b(p)217 2542 y Fy(:=)278 2470 y FB(\( )375 2514 y Fw(n)53 b(n)10 b Fv(\000)h Fy(1)41 b Fv(\001)8 b(\001)g(\001)40 b Fw(n)10 b Fv(\000)g Fy(2)p Fw(p)g Fy(+)h(2)41 b Fw(n)10 b Fv(\000)h Fy(2)p Fw(p)e Fy(+)i(1)48 b Fw(k)h Fv(\001)8 b(\001)g(\001)48 b Fy(1)365 2570 y Fw(\013)394 2577 y Ft(1)485 2570 y Fw(\014)511 2577 y Ft(1)602 2570 y Fv(\001)8 b(\001)g(\001)118 b Fw(\013)804 2577 y Fp(p)1025 2570 y Fw(\014)1051 2577 y Fp(p)1193 2570 y Fw(j)1212 2577 y Ft(1)1273 2570 y Fw(:)8 b(:)g(:)40 b(j)1387 2577 y Fp(k)1428 2470 y FB(!)1474 2542 y Fv(2)13 b Fx(S)1555 2549 y Fp(n)329 2618 y FB(\014)329 2643 y(\014)329 2668 y(\014)368 2667 y Fw(\013)397 2674 y Ft(1)430 2667 y Fw(>)g(:)8 b(:)g(:)j(>)i(\013)621 2674 y Fp(p)648 2667 y Fw(;)20 b(\013)710 2674 y Fp(l)736 2667 y Fw(>)13 b(\014)810 2674 y Fp(l)830 2667 y Fw(;)20 b(l)13 b Fy(=)g(1)p Fw(;)8 b(:)g(:)g(:)t(;)g(p)45 b Fy(and)g Fw(n)13 b Fv(\025)g Fw(j)1355 2674 y Ft(1)1387 2667 y Fw(>)g(j)1454 2674 y Ft(2)1486 2667 y Fw(>)g(:)8 b(:)g(:)e(j)1614 2674 y Fp(k)1648 2667 y Fv(\025)13 b Fy(1)1719 2620 y FB(o)1761 2667 y Fw(:)-12 2814 y Fy(Note)i(that)f Fx(S)234 2821 y Fp(n;)6 b(p)306 2814 y Fy(con)o(tains)484 2742 y FB( )571 2786 y Fw(n)520 2843 y(n)11 b Fv(\000)f Fy(2)p Fw(p)652 2742 y FB(!)697 2784 y Fy(\(2)p Fw(p)p Fy(\)!)p 697 2804 94 2 v 701 2846 a Fw(p)p Fy(!)e(2)768 2832 y Fp(p)823 2814 y Fy(elemen)o(ts.)930 2967 y(3)p eop %%Page: 4 4 4 3 bop -12 50 a Fq(2.2)16 b(Prop)q(osition)24 b Fs(F)m(or)16 b Fw(f)435 57 y Ft(1)455 50 y Fw(;)8 b(:)g(:)g(:)d(;)j(f)579 57 y Fp(n)614 50 y Fv(2)13 b Fx(h)j Fs(the)h(e)n(quation)167 136 y FB(\020)192 183 y Fw(a)p Fy(\()p Fw(f)256 190 y Fp(n)279 183 y Fy(\))10 b Fv(\001)g Fw(:)e(:)g(:)g Fv(\001)i Fw(a)p Fy(\()p Fw(f)480 190 y Ft(1)499 183 y Fy(\))517 136 y FB(\021)549 183 y Fy(\012)j(=)708 143 y FB(X)671 232 y Fr(\031)d Fn(2)f Fm(S)755 236 y Fl(n;)t(p)664 273 y Fk(0)g Fn(\024)f Fk(2)p Fr(p)i Fn(\024)e Fr(n)803 183 y Fy(\(sgn)f Fw(\031)r Fy(\))979 127 y Fp(p)961 143 y FB(Y)960 235 y Fp(l)p Ft(=1)1028 183 y Fv(h)p Fw(P)f(f)1103 190 y Fp(\013)1126 196 y Fr(l)1148 183 y Fw(;)15 b(P)6 b Fy(\000)p Fw(f)1261 190 y Fp(\014)1281 196 y Fr(l)1296 183 y Fv(i)i Fw(P)e Fy(\000)p Fw(f)1407 190 y Fp(j)1421 195 y Fk(1)1451 183 y Fv(^)k Fw(:)e(:)g(:)h Fv(^)h Fw(P)c Fy(\000)p Fw(f)1680 190 y Fp(j)1694 196 y Fr(k)-12 367 y Fs(holds,)16 b(wher)n(e)h(the)f(indic)n(es)f Fw(\013)502 374 y Fp(l)516 367 y Fw(;)8 b(\014)563 374 y Fp(l)574 367 y Fw(;)g(j)614 374 y Ft(1)633 367 y Fw(;)g(:)g(:)g(:)d(;)j(j)754 374 y Fp(k)791 367 y Fs(ar)n(e)16 b(given)g(in)f(the)i(de\014ntion)f (of)g Fx(S)1399 374 y Fp(n;)6 b(p)1472 367 y Fs(and)17 b(wher)n(e)f(for)h Fw(n)c Fy(=)g(2)p Fw(p)-12 424 y Fs(in)j(the)g(pr)n (e)n(c)n(e)n(ding)f(sum)h(one)g(r)n(eplac)n(es)f(the)i(we)n(dge)f(pr)n (o)n(duct)h(by)f(the)h(vacuum)g Fy(\012)p Fs(.)-12 530 y(Pr)n(o)n(of:)j Fy(See)c(app)q(endix.)p 1857 530 24 24 v 59 622 a(Let)22 b Fw(Z)j Fy(b)q(e)d(the)g(implemen)o(tation)i(on)e Fx(F)f Fy(of)h(the)g(ev)o(en{o)q(ddness)h(automorphism)e(asso)q(ciated) h(to)g(the)-12 678 y(Bogoljub)q(o)o(v)16 b(unitarit)o(y)g Fv(\000)p Fy(1)-10 b FF(l)21 b Fy([4)o(,)16 b(p.)f(76].)21 b(It)16 b(satis\014es)g Fw(Z)h Fy(=)d Fw(Z)1058 662 y Fu(\003)1092 678 y Fy(=)g Fw(Z)1175 662 y Fu(\000)p Ft(1)1238 678 y Fy(and)i(therefore)g(its)g(sp)q(ectral)g(decom-)-12 735 y(p)q(osition)g(is)g(simply)g(giv)o(en)g(b)o(y)791 791 y Fw(Z)g Fy(=)d Fw(E)923 772 y Ft(+)961 791 y Fv(\000)e Fw(E)1044 772 y Fu(\000)1080 791 y Fw(:)745 b Fy(\(4\))-12 875 y(Let)17 b(further)f Fw(X)i Fy(=)c Fw(X)368 882 y Fk(ev)o(en)440 875 y Fy(+)e Fw(X)525 882 y Fk(o)q(dd)592 875 y Fy(b)q(e)17 b(the)g(unique)h(decomp)q(osition)f(of)f(an)o(y)g Fw(X)i Fv(2)d Fy(CAR\()p Fx(h)p Fw(;)8 b Fy(\000\))15 b(in)o(to)i(its)f(ev)o(en)-12 931 y(and)f(o)q(dd)h(parts.)j(The)d (follo)o(wing)g(result)f(will)i(b)q(e)f(used)g(in)g(Section)g(7.)-12 1025 y Fq(2.3)g(Lemma)24 b Fs(L)n(et)15 b Fw(\031)i Fs(b)n(e)e(a)h(F)m (o)n(ck)e(r)n(epr)n(esentation)h(of)g Fy(CAR)q(\()p Fx(h)p Fw(;)8 b Fy(\000\))14 b Fs(and)h Fw(Z)h Fy(=)d Fw(E)1353 1008 y Ft(+)1390 1025 y Fv(\000)c Fw(E)1471 1008 y Fu(\000)1515 1025 y Fs(as)16 b(b)n(efor)n(e.)k(Then)14 b(for)-12 1081 y(any)i Fw(X)g Fv(2)d Fy(CAR\()p Fx(h)p Fw(;)8 b Fy(\000\))15 b Fs(we)i(have)542 1183 y Fw(E)579 1165 y Ft(+)608 1183 y Fw(\031)r Fy(\()p Fw(X)692 1190 y Fk(ev)o(en)752 1183 y Fy(\))p Fw(E)807 1165 y Fu(\000)877 1183 y Fy(=)42 b Fw(E)991 1165 y Fu(\000)1020 1183 y Fw(\031)r Fy(\()p Fw(X)1104 1190 y Fk(ev)o(en)1164 1183 y Fy(\))p Fw(E)1219 1165 y Ft(+)1260 1183 y Fy(=)13 b(0)552 1252 y Fw(E)589 1234 y Ft(+)618 1252 y Fw(\031)r Fy(\()p Fw(X)702 1259 y Fk(o)q(dd)752 1252 y Fy(\))p Fw(E)807 1234 y Ft(+)877 1252 y Fy(=)42 b Fw(E)991 1234 y Fu(\000)1020 1252 y Fw(\031)r Fy(\()p Fw(X)1104 1259 y Fk(o)q(dd)1154 1252 y Fy(\))p Fw(E)1209 1234 y Fu(\000)1250 1252 y Fy(=)13 b(0)8 b Fw(:)-12 1354 y Fs(Pr)n(o)n(of:)19 b Fy(Recall)d(that)e Fw(Z)392 1361 y Ft(0)412 1354 y Fw(\031)r Fy(\()p Fw(X)496 1361 y Fk(ev)o(en)556 1354 y Fy(\))e(=)h Fw(\031)r Fy(\()p Fw(X)718 1361 y Fk(ev)o(en)778 1354 y Fy(\))p Fw(Z)827 1361 y Ft(0)847 1354 y Fy(.)19 b(Multiplying)e(from)c(the)h(left)h(b)o (y)f Fw(E)1487 1338 y Ft(+)1530 1354 y Fy(and)g(from)g(the)g(righ)o(t) -12 1411 y(b)o(y)h Fw(E)88 1394 y Fu(\000)132 1411 y Fy(w)o(e)g(get)590 1467 y Fw(E)627 1449 y Ft(+)656 1467 y Fw(\031)r Fy(\()p Fw(X)740 1474 y Fk(ev)o(en)800 1467 y Fy(\))p Fw(E)855 1449 y Fu(\000)896 1467 y Fy(=)e Fv(\000)p Fw(E)1016 1449 y Ft(+)1045 1467 y Fw(\031)r Fy(\()p Fw(X)1129 1474 y Fk(ev)o(en)1190 1467 y Fy(\))p Fw(E)1245 1449 y Fu(\000)1281 1467 y Fw(;)-12 1551 y Fy(whic)o(h)j(implies)h(the)f (\014rst)e(t)o(w)o(o)g(eqs.)20 b(Similarly)d(w)o(e)e(obtain)h(the)f (eqs.)g(corresp)q(onding)h(to)f(the)g(o)q(dd)h(part.)p 1857 1551 V -12 1708 a Fj(2.1)56 b(Direct)17 b(sums)h(and)h(tensor)f (pro)r(ducts)-12 1794 y Fy(Let)i(\()p Fx(h)116 1801 y Fp(k)136 1794 y Fw(;)8 b Fy(\000)185 1801 y Fp(k)206 1794 y Fy(\),)20 b Fw(k)h Fy(=)g(0)p Fw(;)8 b Fy(1,)18 b(b)q(e)j(t)o(w)o(o)d(Hilb)q(ert)j(spaces)f(with)g(an)o(ti{unitary)f (in)o(v)o(olutions)i(and)f(let)g Fw(P)1718 1801 y Fp(k)1759 1794 y Fy(b)q(e)g(t)o(w)o(o)-12 1850 y(basis)h(pro)s(jections)f(for)f (the)i(corresp)q(onding)g(spaces.)36 b(W)l(e)20 b(denote)h(b)o(y)f Fw(\031)1301 1857 y Fp(k)1342 1850 y Fy(the)h(F)l(o)q(c)o(k)f(represen) o(tations)g(of)-12 1907 y(CAR\()p Fx(h)130 1914 y Fp(k)151 1907 y Fw(;)8 b Fy(\000)200 1914 y Fp(k)221 1907 y Fy(\))19 b(on)g(the)h(F)l(o)q(c)o(k)f(spaces)g Fx(F)691 1914 y Fp(k)712 1907 y Fy(.)33 b Fw(Z)789 1914 y Fp(k)829 1907 y Fy(are)19 b(the)h(implemen)o(ters)g(of)f(the)g(resp)q(ectiv)o(e)i(ev) o(en{o)q(ddness)-12 1963 y(automorphisms,)14 b Fw(k)g Fy(=)f(0)p Fw(;)8 b Fy(1.)19 b(Motiv)m(ated)c(b)o(y)g([14)o(,)g(p.)g (82])f(\(cf.)h(also)g([19)o(,)g(p.)g(219]\))e(w)o(e)i(consider:)-12 2057 y Fq(2.4)h(Prop)q(osition)24 b Fs(With)c(the)g(pr)n(e)n(c)n(e)n (ding)d(notation)j(put)g Fx(h)d Fy(:=)h Fx(h)1120 2064 y Ft(0)1152 2057 y Fv(\010)13 b Fx(h)1224 2064 y Ft(1)1262 2057 y Fs(and)19 b Fy(\000)g(:=)f(\000)1494 2064 y Ft(0)1526 2057 y Fv(\010)13 b Fy(\000)1602 2064 y Ft(1)1641 2057 y Fs(and)20 b(c)n(onsider)-12 2113 y Fy(CAR\()p Fx(h)p Fw(;)8 b Fy(\000\))p Fs(.)20 b(The)c(r)n(epr)n(esentation)f(de\014ne)n (d)g(on)h Fx(F)866 2120 y Ft(0)896 2113 y Fv(\012)10 b Fx(F)969 2120 y Ft(1)1005 2113 y Fs(by)249 2223 y Fw(\031)277 2176 y FB(\020)301 2223 y Fw(a)p Fy(\()p Fw(f)365 2230 y Ft(0)395 2223 y Fv(\010)g Fw(f)462 2230 y Ft(1)482 2223 y Fy(\))500 2176 y FB(\021)538 2223 y Fy(:=)i Fw(\031)624 2230 y Ft(0)644 2176 y FB(\020)669 2223 y Fw(a)p Fy(\()p Fw(f)733 2230 y Ft(0)752 2223 y Fy(\))770 2176 y FB(\021)805 2223 y Fv(\012)e Fy(1)-10 b FF(l)15 b Fy(+)10 b Fw(Z)966 2230 y Ft(0)996 2223 y Fv(\012)h Fw(\031)1068 2230 y Ft(1)1087 2176 y FB(\020)1112 2223 y Fw(a)p Fy(\()p Fw(f)1176 2230 y Ft(1)1196 2223 y Fy(\))1214 2176 y FB(\021)1246 2223 y Fw(;)32 b(f)1313 2230 y Fp(k)1347 2223 y Fv(2)13 b Fx(h)1414 2230 y Fp(k)1443 2223 y Fw(;)8 b(k)13 b Fy(=)g(0)p Fw(;)8 b Fy(1)g Fw(;)201 b Fy(\(5\))-12 2333 y Fs(is)12 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(osition)g(w)o(e)f(can)h(use)f(the)h(results)f(in)h(Mo)q(dular)g (Theory)f([25)o(,)g(12)o(])g(for)-12 1581 y(the)j(pair)g(\()p Fv(M)p Fy(\()p Fx(q)p Fy(\))p Fw(;)8 b Fy(\012\).)27 b(These)19 b(tec)o(hniques)h(will)g(b)q(e)f(essen)o(tially)h(used)g(in) f(the)g(next)f(t)o(w)o(o)g(sections)h(for)f(the)-12 1637 y(pro)q(of)d(of)g(the)g(t)o(wisted)g(dualit)o(y)h(prop)q(ert)o(y)f(in)h (the)f(generic)h(p)q(osition)g(con)o(text.)-12 1744 y Fq(3.3)g(Lemma)24 b Fs(L)n(et)e Fx(q)j Fv(\032)g Fx(h)e Fs(b)n(e)f(a)h(close)n(d)f Fy(\000)p Fs({invariant)h(subsp)n(ac)n(e.)40 b(Then)22 b(the)h(fol)r(lowing)f(c)n(onditions)g(ar)n(e)-12 1800 y(e)n(quivalent:)28 1894 y(\(i\))g Fw(q)14 b Fv(2)f Fx(q)k Fs(and)f Fw(P)6 b(q)15 b Fy(=)e(0)j Fs(implies)f Fw(q)g Fy(=)e(0)p Fs(.)14 1988 y(\(ii\))22 b Fw(P)6 b Fy(\()p Fx(q)177 1971 y Fu(?)206 1988 y Fy(\))16 b Fs(is)g(a)h(dense)e (submanifold)h(of)h Fx(p)p Fs(.)0 2081 y(\(iii\))22 b Fx(q)10 b Fv(\\)g Fx(p)j Fy(=)g Fv(f)p Fy(0)p Fv(g)p Fs(.)-12 2188 y(Pr)n(o)n(of:)23 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b(vacuum)h(ve)n(ctor)f Fy(\012)g Fs(is)g(sep)n(ar)n(ating)f(for)i Fv(M)p Fy(\()p Fx(q)p Fy(\))28 b Fs(i\013)h Fw(P)6 b Fy(\()p Fx(q)1169 2613 y Fu(?)1199 2630 y Fy(\))16 b Fs(is)f(a)i(dense)e(submanifold)h(of)h Fx(p)p Fs(.)930 2967 y Fy(7)p eop %%Page: 8 8 8 7 bop -12 50 a Fs(Pr)n(o)n(of:)19 b Fy(\(i\))13 b(W)l(e)g(assume)h (that)e(\012)h(is)h(cyclic)h(for)e Fv(M)p Fy(\()p Fx(q)p Fy(\))f(and)i(let)f Fw(p)g Fv(2)g Fx(p)g Fy(b)q(e)h(a)f(v)o(ector)f (satisfying)i Fw(p)f Fv(?)g Fw(P)6 b Fx(q)p Fy(.)19 b(F)l(rom)-12 107 y(this,)c(from)g(Prop)q(osition)g(2.2)g(and)g(from)f(the)i (structure)f(of)f(the)i(F)l(o)q(c)o(k)f(space)g Fx(F)g Fy(\(recall)h(Eq.)f(\(3\)\))f(w)o(e)h(ha)o(v)o(e)322 209 y Fw(p)42 b Fv(?)g Fy(span)7 b Fv(f)p Fw(a)p Fy(\()p Fw(q)647 216 y Ft(1)667 209 y Fy(\))j Fv(\001)f Fw(:)f(:)g(:)g Fv(\001)i Fw(a)p Fy(\()p Fw(q)865 216 y Fp(n)889 209 y Fy(\)\012)i Fv(j)g Fw(q)997 216 y Ft(1)1017 209 y Fw(;)c(:)g(:)g(:)d (q)1118 216 y Fp(n)1154 209 y Fv(2)13 b Fx(q)8 b Fw(;)20 b(n)13 b 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Fv(\032)i Fx(F)8 b Fw(;)-12 802 y Fy(whic)o(h)16 b(imply)h(that)d(\012) h(is)h(cyclic)h(for)d Fv(M)p Fy(\()p Fx(q)p Fy(\).)59 858 y(\(ii\))i(Supp)q(ose)g(no)o(w)g(that)e(\012)i(is)g(a)f(separating) h(v)o(ector)f(for)g Fv(M)p Fy(\()p Fx(q)p Fy(\).)20 b(W)l(e)c(sho)o(w)f (that)g(this)h(implies)i(part)d(\(i\))-12 915 y(of)f(Lemma)g(3.3.)k (So,)c(let)h Fw(q)f Fv(2)f Fx(q)i Fy(satisfy)e Fw(P)6 b(q)15 b Fy(=)e(0.)20 b(Since)15 b(0)e(=)g Fw(P)6 b(q)15 b Fy(=)e Fw(a)p Fy(\(\000)p Fw(q)r Fy(\)\012)g(and)i(since)g(\012)f(is) g(separating)g(for)-12 971 y Fv(M)p Fy(\()p Fx(q)p Fy(\))g(w)o(e)h(m)o (ust)g(ha)o(v)o(e)g Fw(a)p Fy(\(\000)p Fw(q)r Fy(\))d(=)h(0,)h(whic)o (h)i(implies)i Fw(q)c Fy(=)f(0.)59 1028 y(Finally)l(,)18 b(assume)e(that)g Fw(P)6 b Fy(\()p Fx(q)557 1011 y Fu(?)587 1028 y Fy(\))16 b(is)h(a)g(dense)g(submanifold)h(of)e Fx(p)p Fy(.)24 b(By)17 b(part)f(\(i\))g(of)h(the)f(presen)o(t)h (theorem)-12 1084 y(applied)k(to)d Fx(q)232 1068 y Fu(?)262 1084 y Fy(,)h(whic)o(h)h(is)g(also)f(\000{in)o(v)m(arian)o(t,)h(w)o(e)f (kno)o(w)f(that)h(\012)f(is)i(cyclic)h(for)d Fv(M)p Fy(\()p Fx(q)1526 1068 y Fu(?)1555 1084 y Fy(\).)31 b(F)l(urther,)20 b(since)-3 1127 y FB(e)-12 1140 y Fw(Z)s Fy(\012)13 b(=)h(\012)h(w)o(e) h(ha)o(v)o(e)f(that)g(\012)g(is)h(also)g(cyclic)h(for)828 1127 y FB(e)818 1140 y Fw(Z)11 b Fv(M)p Fy(\()p Fx(q)955 1124 y Fu(?)984 1140 y Fy(\))1019 1127 y FB(e)1009 1140 y Fw(Z)1043 1124 y Fu(\003)1079 1140 y Fy(and)16 b(consequen)o(tly)g(b) o(y)g(Prop)q(osition)g(2.6)f(also)-12 1197 y(for)g Fv(M)p Fy(\()p Fx(q)p Fy(\))171 1180 y Fu(0)194 1197 y Fv(\023)251 1184 y FB(e)242 1197 y Fw(Z)c Fv(M)p Fy(\()p Fx(q)379 1180 y Fu(?)408 1197 y Fy(\))442 1184 y FB(e)433 1197 y Fw(Z)467 1180 y Fu(\003)487 1197 y Fy(.)20 b(This)c(sho)o(ws)e(that)h (\012)g(is)g(separating)g(for)g Fv(M)p Fy(\()p Fx(q)p Fy(\).)p 1857 1197 24 24 v 59 1289 a(By)g(the)h(preceding)h(result)f(w) o(e)g(kno)o(w)f(that)g(if)h Fx(q)g Fy(is)g(a)f(closed)i(\000{in)o(v)m (arian)o(t)f(subspace)g(of)g Fx(h)f Fy(where)h Fw(P)6 b Fx(q)16 b Fy(as)-12 1345 y(w)o(ell)f(as)e Fw(P)6 b Fy(\()p Fx(q)208 1329 y Fu(?)238 1345 y Fy(\))13 b(are)g(dense)i (submanifolds)g(of)e Fx(p)p Fy(,)g(then)h(the)g(Mo)q(dular)f(Theory)h (is)g(w)o(ell{de\014ned)i(for)d(the)g(pairs)-12 1402 y(\()p Fv(M)p Fy(\()p Fx(q)p Fy(\))p Fw(;)8 b Fy(\012\))16 b(and)j(\()p Fv(M)p Fy(\()p Fx(q)412 1385 y Fu(?)441 1402 y Fy(\))p Fw(;)8 b Fy(\012\).)29 b(Denote)18 b(b)o(y)h Fw(S)i Fy(and)e Fw(T)25 b Fy(the)18 b(T)l(omita)h(op)q(erators)f (corresp)q(onding)h(to)f(these)-12 1458 y(pairs)d(resp)q(ectiv)o(ely)l (.)22 b(W)l(e)15 b(will)i(next)f(study)f(their)h(action)f(on)g(the)g (submanifolds)i Fw(P)6 b Fx(q)16 b Fy(and)f Fw(P)6 b Fy(\()p Fx(q)1650 1442 y Fu(?)1680 1458 y Fy(\).)-12 1564 y Fq(3.5)16 b(Lemma)24 b Fs(With)17 b(the)g(pr)n(e)n(c)n(e)n(ding) e(notation)h(we)g(have)h(for)f Fw(q)f Fv(2)e Fx(q)j Fs(and)h Fw(q)1281 1548 y Fu(?)1323 1564 y Fv(2)c Fx(q)1388 1548 y Fu(?)1417 1564 y Fs(:)28 1658 y(\(i\))22 b Fw(S)s Fy(\()p Fw(P)6 b(q)r Fy(\))11 b(=)i Fw(P)6 b Fy(\000)p Fw(q)r Fs(.)14 1752 y(\(ii\))22 b Fw(T)6 b Fy(\()p Fw(P)g(q)210 1736 y Fu(?)239 1752 y Fy(\))12 b(=)h Fw(P)6 b Fy(\000)p Fw(q)402 1736 y Fu(?)445 1752 y Fy(=)13 b Fv(\000)p Fw(S)559 1736 y Fu(\003)579 1752 y Fy(\()p Fw(P)6 b(q)654 1736 y Fu(?)683 1752 y Fy(\))p Fs(.)-12 1858 y(Pr)n(o)n(of:)20 b Fy(The)15 b(\014rst)g(t)o(w)o(o)f(equations)h(follo)o(w)h(b)o(y)f (direct)h(computation:)445 1966 y Fw(S)s Fy(\()p Fw(P)6 b(q)r Fy(\))12 b(=)h Fw(S)660 1919 y FB(\020)684 1966 y Fw(a)p Fy(\(\000)p Fw(q)r Fy(\))8 b(\012)835 1919 y FB(\021)872 1966 y Fy(=)13 b Fw(a)p Fy(\(\000)p Fw(q)r Fy(\))1030 1947 y Fu(\003)1057 1966 y Fy(\012)f(=)h Fw(a)p Fy(\()p Fw(q)r Fy(\))8 b(\012)k(=)h Fw(P)6 b Fy(\000)p Fw(q)k(;)-12 2073 y Fy(and)15 b(similarly)i(for)e Fw(T)6 b Fy(.)19 b(T)l(o)c(pro)o(v)o(e)g(the)g(last)g(equation)g(recall)i (that)d Fw(S)1187 2057 y Fu(\003)1221 2073 y Fy(is)i(actually)g(the)f (T)l(omita)g(op)q(erator)f(of)-12 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Fw(i)8 b(S)515 2461 y Fu(\003)533 2432 y FB(\020)568 2466 y(e)558 2480 y Fw(Z)s(a)p Fy(\(\000)p Fw(q)684 2461 y Fu(?)714 2480 y Fy(\))741 2466 y FB(e)732 2480 y Fw(Z)766 2461 y Fu(\003)793 2480 y Fy(\012)826 2432 y FB(\021)863 2480 y Fy(=)13 b Fv(\000)p Fw(i)979 2466 y FB(e)970 2480 y Fw(Z)s(a)p Fy(\()p Fw(q)1068 2461 y Fu(?)1097 2480 y Fy(\))1125 2466 y FB(e)1115 2480 y Fw(Z)1149 2461 y Fu(\003)1177 2480 y Fy(\012)f(=)h Fv(\000)p Fw(P)6 b Fy(\000)p Fw(q)1390 2461 y Fu(?)1433 2480 y Fy(=)13 b Fv(\000)p Fw(T)6 b Fy(\()p Fw(P)g(q)1624 2461 y Fu(?)1654 2480 y Fy(\))i Fw(;)-12 2587 y Fy(and)15 b(the)h(pro)q(of)e(is)i (concluded.)p 1857 2587 24 24 v 930 2967 a(8)p eop %%Page: 9 9 9 8 bop -12 50 a Fz(4)67 b(P)n(airs)23 b(of)e(pro)t(jections)h(in)i (generic)f(p)r(osition)g(and)g(the)f(mapping)i Ff(')-12 152 y Fy(In)11 b(this)g(section)f(w)o(e)g(will)i(consider)f(the)g (mathematically)g(ric)o(hest)f(situation)h(whic)o(h)g(app)q(ears)g (when)f(the)h(closed)-12 208 y(subspaces)18 b Fx(p)f Fy(and)g Fx(q)g Fy(are)g(in)h(generic)g(p)q(osition.)27 b(Let)17 b Fw(P)24 b Fy(and)17 b Fw(Q)g Fy(b)q(e)h(the)g(corresp)q (onding)g(orthopro)s(jections)-12 265 y(satisfying)13 b(as)f(usual)h(the)f(relations)h(\000)p Fw(P)6 b Fy(\000)14 b(=)f(1)-11 b FF(l)10 b Fv(\000)t Fw(P)20 b Fy(=)13 b Fw(P)972 248 y Fu(?)1014 265 y Fy(and)g Fw(Q)p Fy(\000)g(=)g(\000)p Fw(Q)p Fy(.)19 b(Motiv)m(ated)12 b(b)o(y)h(Prop)q(osition)g(3.4)-12 321 y(w)o(e)i(will)i(also)e(assume)g(here)h(that)546 420 y Fx(p)9 b Fv(\\)i Fx(q)i Fy(=)g Fv(f)p Fy(0)p Fv(g)f Fy(=)g Fx(p)e Fv(\\)h Fx(q)926 401 y Fu(?)968 420 y Fw(;)53 b Fy(where)13 b Fx(p)f Fy(:=)h Fw(P)6 b Fx(h)i Fw(;)476 b Fy(\(10\))-12 518 y(whic)o(h)16 b(directly)h(implies)g(using)f(the)f (basis)h(pro)s(jection)f(prop)q(ert)o(y:)493 616 y Fx(p)516 598 y Fu(?)555 616 y Fv(\\)c Fx(q)i Fy(=)f Fv(f)p Fy(0)p Fv(g)g Fy(=)h Fx(p)830 598 y Fu(?)870 616 y Fv(\\)d Fx(q)932 598 y Fu(?)974 616 y Fw(;)53 b Fy(where)13 b Fx(p)1192 598 y Fu(?)1234 616 y Fy(=)g Fw(P)1317 598 y Fu(?)1347 616 y Fx(h)8 b Fw(:)423 b Fy(\(11\))-12 715 y(Th)o(us)14 b(in)g(the)g(notation)f(of)h(Section)g(3)g(w)o(e)f(ha)o(v)o(e)h(the)g (extremal)f(case)h(where)g Fx(p)g Fy(and)g Fx(q)g Fy(are)f(already)h (in)h(generic)-12 771 y(p)q(osition)h(and)g(the)f(Halmos)g(decomp)q (osition)h(\(9\))f(is)g(trivial)i(in)f(the)f(sense)h(that)e Fx(h)1415 778 y Ft(0)1447 771 y Fy(=)f Fv(f)p Fy(0)p Fv(g)p Fy(.)59 828 y(Note)h(that)f(the)i(prop)q(erties)g(lik)o(e)h Fx(p)f Fv(\\)i Fx(q)c Fy(=)g Fv(f)p Fy(0)p Fv(g)g Fy(can)i(b)q(e)g (also)f(expressed)i(b)o(y)e(the)h(corresp)q(onding)g(pro)s(jec-)-12 884 y(tions)g Fw(P)22 b Fy(and)15 b Fw(Q)p Fy(,)g(b)q(ecause)h(the)g (orthopro)s(jection)e(on)o(to)g(the)h(in)o(tersection)h Fx(p)10 b Fv(\\)h Fx(q)k Fy(is)h(giv)o(en)f(b)o(y)640 983 y(s)p Fv(\000)7 b Fy(lim)656 1011 y Fp(n)p Fu(!1)771 983 y Fy(\()p Fw(P)f(Q)p Fy(\))878 964 y Fp(n)914 983 y Fy(=)13 b(s)p Fv(\000)8 b Fy(lim)978 1011 y Fp(n)p Fu(!1)1094 983 y Fy(\()p Fw(QP)e Fy(\))1201 964 y Fp(n)1231 983 y Fw(:)-12 1096 y Fy(Note)12 b(further)f(that)g(b)o(y)h(Lemma)g (3.3)f(the)h(in)o(tersection)h(assumptions)f(in)h(\(10\))e(are)g(equiv) m(alen)o(t)j(to)d(the)h(densit)o(y)-12 1152 y(conditions)k(in)g(Prop)q (osition)g(3.4.)-12 1255 y Fq(4.1)g(Rema)o(rk)23 b Fy(The)14 b(follo)o(wing)g(useful)g(densit)o(y)g(statemen)o(ts)e(are)h(immediate) h(consequences)h(of)d(the)i(assump-)-12 1311 y(tion)j(that)g Fx(p)g Fy(and)g Fx(q)g Fy(are)g(in)h(generic)g(p)q(osition.)27 b(If)18 b Fx(r)d Fv(\022)h Fx(q)i Fy(\(or)e Fx(r)g Fv(\022)g Fx(q)1193 1295 y Fu(?)1223 1311 y Fy(\))g(is)i(a)f(dense)h(linear)g (submanifold)h(in)-12 1368 y Fx(q)e Fy(\(resp.)g(in)h Fx(q)233 1351 y Fu(?)262 1368 y Fy(\),)f(then)h Fw(P)6 b Fx(r)17 b Fy(is)h(dense)g(in)f Fx(p)g Fy(and)h Fw(P)880 1351 y Fu(?)910 1368 y Fx(r)e Fy(is)i(dense)g(in)g Fx(p)1196 1351 y Fu(?)1225 1368 y Fy(.)26 b(The)17 b(same)g(holds)h(if)f Fw(Q)g Fy(and)g Fw(P)24 b Fy(are)-12 1424 y(in)o(terc)o(hanged.)c(Th)o (us)c(w)o(e)e(ha)o(v)o(e)h(for)g(example)h(that)e Fw(Q)p Fx(p)965 1408 y Fu(?)1010 1424 y Fy(is)h(dense)h(in)g Fx(q)p Fy(,)f Fw(P)6 b(Q)p Fx(p)16 b Fy(is)g(dense)f(in)i Fx(p)d Fy(etc.)59 1527 y(W)l(e)g(will)i(b)q(egin)f(next)f(a)g (systematic)g(analysis)h(of)e(a)h(mapping)h Fw(')f Fy(that)f(can)i(b)q (e)f(naturally)h(de\014ned)h(in)f(the)-12 1583 y(presen)o(t)g(con)o (text.)k(Put)487 1687 y Fx(H)520 1694 y Fp(')587 1687 y Fy(:=)676 1640 y FB(n)704 1687 y Fy(\()p Fw(q)r(;)8 b(q)787 1668 y Fu(?)815 1687 y Fy(\))13 b Fv(2)f Fx(q)f Fv(\002)f Fx(q)988 1668 y Fu(?)1031 1687 y Fv(j)i Fw(P)1091 1668 y Fu(?)1121 1687 y Fy(\()p Fw(q)g Fy(+)e Fw(q)1238 1668 y Fu(?)1267 1687 y Fy(\))j(=)g(0)1369 1640 y FB(o)492 1778 y Fx(H)525 1785 y Fp(\032)587 1778 y Fy(:=)676 1731 y FB(n)704 1778 y Fy(\()p Fw(q)r(;)8 b(q)787 1760 y Fu(?)815 1778 y Fy(\))13 b Fv(2)f Fx(q)f Fv(\002)f Fx(q)988 1760 y Fu(?)1031 1778 y Fv(j)i Fw(P)6 b(q)15 b Fy(=)e Fw(P)6 b(q)1231 1760 y Fu(?)1261 1731 y FB(o)1296 1778 y Fw(:)-12 1882 y Fq(4.2)16 b(Lemma)24 b Fs(The)19 b(sets)g Fx(H)465 1889 y Fp(')509 1882 y Fs(and)h Fx(H)634 1889 y Fp(\032)673 1882 y Fs(ar)n(e)f(gr)n(aphs)g(of)h(line)n(ar,)f(inje)n(ctive)f(and)i (close)n(d)e(mappings)h Fw(';)8 b(\032)p Fy(:)g Fx(q)16 b Fv(!)-12 1939 y Fx(q)10 1922 y Fu(?)62 1939 y Fs(with)22 b(dense)g(domains)g(and)g(dense)f(images.)38 b(The)22 b(gr)n(aphs)30 b Fy(gra)7 b Fw(')22 b Fy(=)i Fx(H)1367 1946 y Fp(')1414 1939 y Fs(and)30 b Fy(gra)6 b Fw(\032)23 b Fy(=)h Fx(H)1725 1946 y Fp(\032)1767 1939 y Fs(c)n(an)d(b)n(e)-12 1995 y(p)n(ar)n(ametrize)n(d)16 b(by)g Fx(p)g Fs(r)n(esp.)g Fx(p)486 1978 y Fu(?)532 1995 y Fs(as)541 2099 y Fy(gra)7 b Fw(')41 b Fy(:=)772 2052 y FB(n)800 2099 y Fy(\()p Fw(Qp;)8 b(Q)934 2080 y Fu(?)962 2099 y Fw(p)p Fy(\))k Fv(j)g Fw(p)h Fv(2)g Fx(p)1142 2052 y FB(o)1815 2099 y Fy(\(12\))547 2190 y(gra)7 b Fw(\032)41 b Fy(:=)772 2143 y FB(n)800 2190 y Fy(\()p Fw(Qp)877 2171 y Fu(?)906 2190 y Fw(;)8 b Fv(\000)p Fw(Q)998 2171 y Fu(?)1027 2190 y Fw(p)1050 2171 y Fu(?)1079 2190 y Fy(\))13 b Fv(j)f Fw(p)1158 2171 y Fu(?)1200 2190 y Fv(2)h Fx(p)1266 2171 y Fu(?)1295 2143 y FB(o)1330 2190 y Fw(;)-12 2300 y Fs(wher)n(e)j(the)h (domains)f(and)g(images)g(ar)n(e)h(given)e(explicitly.)20 b(Mor)n(e)n(over,)c(the)h(e)n(quation)f Fw(\032)1512 2284 y Fu(\000)p Ft(1)1571 2300 y Fy(=)d Fw(')1649 2284 y Fu(\003)1685 2300 y Fs(holds.)-12 2391 y(Pr)n(o)n(of:)24 b Fy(W)l(e)18 b(consider)g(\014rst)f(the)h(mapping)g Fw(')f Fy(and)g(one)h(can)g(similarly)h(argue)e(for)g Fw(\032)p Fy(.)26 b(First)17 b(note)g(that)g(the)-12 2447 y(assignmen)o(t)j Fw(')p Fy(\()p Fw(q)r Fy(\))g(:=)i Fw(q)427 2431 y Fu(?)476 2447 y Fy(if)f Fw(P)558 2431 y Fu(?)588 2447 y Fy(\()p Fw(q)15 b Fy(+)f Fw(q)712 2431 y Fu(?)742 2447 y Fy(\))21 b(=)g(0)f(is)h(a)f(w)o(ell{de\014ned)j (linear)e(map.)36 b(Indeed,)22 b(if)f(\()p Fw(q)r(;)8 b(q)1798 2431 y Fu(?)1796 2459 y Ft(1)1826 2447 y Fy(\))21 b Fv(2)-12 2504 y Fx(H)21 2511 y Fp(')61 2504 y Fv(3)15 b Fy(\()p Fw(q)r(;)8 b(q)189 2487 y Fu(?)187 2515 y Ft(2)217 2504 y Fy(\),)16 b(then)h Fw(q)391 2487 y Fu(?)389 2515 y Ft(1)436 2504 y Fy(=)e Fw(q)508 2487 y Fu(?)506 2515 y Ft(2)537 2504 y Fy(,)i(b)q(ecause)g(in)h(this)f(case)f Fw(q)1000 2487 y Fu(?)998 2515 y Ft(1)1041 2504 y Fv(\000)11 b Fw(q)1109 2487 y Fu(?)1107 2515 y Ft(2)1154 2504 y Fv(2)k Fx(q)1221 2487 y Fu(?)1262 2504 y Fv(\\)c Fx(p)k Fy(=)g Fv(f)p Fy(0)p Fv(g)p Fy(.)23 b(Injectivit)o(y)18 b(is)f(pro)o(v)o(ed)-12 2560 y(analogously)l(.)59 2617 y(Next)22 b(w)o(e)f(sho)o(w)h(Eq.)g(\(12\).)39 b(Let)23 b Fx(p)h Fv(3)g Fw(p)g Fy(=)h Fw(Qp)14 b Fy(+)i Fw(Q)1051 2600 y Fu(?)1080 2617 y Fw(p)p Fy(,)23 b(so)f(that)g Fw(P)1343 2600 y Fu(?)1373 2617 y Fy(\()p Fw(Qp)14 b Fy(+)h Fw(Q)1550 2600 y Fu(?)1579 2617 y Fw(p)p Fy(\))24 b(=)h(0)d(and)g(w)o(e)-12 2673 y(ha)o(v)o(e)d Fw(Q)p Fx(p)g Fv(\022)h Fy(dom)8 b Fw(')19 b Fy(as)g(w)o(ell)i(as)e Fw(Q)626 2657 y Fu(?)655 2673 y Fx(p)h Fv(\022)g Fy(ima)7 b Fw(')p Fy(.)33 b(T)l(o)19 b(sho)o(w)g(the)g(rev)o(erse)h(inclusions)h (tak)o(e)e Fw(q)1670 2680 y Ft(0)1710 2673 y Fv(2)h Fy(dom)7 b Fw(')p Fy(,)-12 2730 y(i.e.)21 b Fw(P)102 2713 y Fu(?)132 2730 y Fy(\()p Fw(q)170 2737 y Ft(0)204 2730 y Fy(+)14 b Fw(q)275 2713 y Fu(?)273 2741 y Ft(0)304 2730 y Fy(\))22 b(=)h(0)d(for)h(some)f Fw(q)662 2713 y Fu(?)660 2741 y Ft(0)714 2730 y Fv(2)i Fx(q)788 2713 y Fu(?)818 2730 y Fy(.)37 b(But)21 b(this)g(implies)i(that)e Fw(q)1343 2737 y Ft(0)1385 2730 y Fy(=)h Fw(Q)p Fy(\()p Fw(q)1516 2737 y Ft(0)1550 2730 y Fy(+)14 b Fw(q)1621 2713 y Fu(?)1619 2741 y Ft(0)1651 2730 y Fy(\))21 b Fv(2)i Fw(Q)p Fx(p)d Fy(and)-12 2786 y Fw(q)10 2770 y Fu(?)8 2798 y Ft(0)52 2786 y Fy(=)13 b Fw(Q)136 2770 y Fu(?)165 2786 y Fy(\()p Fw(q)203 2793 y Ft(0)230 2786 y Fy(+)7 b Fw(q)294 2770 y Fu(?)292 2798 y Ft(0)324 2786 y Fy(\))13 b Fv(2)f Fw(Q)433 2770 y Fu(?)463 2786 y Fx(p)h Fy(and)h(Eq.)g(\(12\))e(is)i(pro)o(v)o (ed.)19 b(Note)14 b(that)f(b)o(y)g(the)h(preceding)h(remark)e(the)h (domain)-12 2843 y(and)h(image)h(of)e Fw(')h Fy(are)g(dense)h(in)g Fx(q)g Fy(resp.)f Fx(q)726 2826 y Fu(?)770 2843 y Fy(and)h(it)f(is)h (easy)f(to)g(see)g(that)g(gra)6 b Fw(')15 b Fy(is)h(closed.)930 2967 y(9)p eop %%Page: 10 10 10 9 bop 59 50 a Fy(Finally)l(,)16 b(it)f(remains)h(to)f(sho)o(w)f (that)h Fw(\032)728 34 y Fu(\000)p Ft(1)787 50 y Fy(=)e Fw(')865 34 y Fu(\003)884 50 y Fy(.)20 b(Recall)d(\014rst)e(that)555 149 y(gra)7 b Fw(\032)650 131 y Fu(\000)p Ft(1)710 149 y Fy(=)758 102 y FB(n)785 149 y Fy(\()p Fw(Q)839 131 y Fu(?)868 149 y Fw(p)891 131 y Fu(?)921 149 y Fw(;)h Fv(\000)p Fw(Qp)1036 131 y Fu(?)1065 149 y Fy(\))k Fv(j)g Fw(p)1143 131 y Fu(?)1185 149 y Fv(2)h Fx(p)1251 131 y Fu(?)1280 102 y FB(o)1316 149 y Fw(:)-12 257 y Fy(Using)j(the)f (isometric)g(mapping)h Fw(U)5 b Fy(:)22 b Fx(q)10 b Fv(\002)h Fx(q)746 240 y Fu(?)788 257 y Fv(!)i Fx(q)868 240 y Fu(?)908 257 y Fv(\002)d Fx(q)15 b Fy(giv)o(en)h(b)o(y)f Fw(U)5 b Fy(\()p Fw(q)r(;)j(q)1291 240 y Fu(?)1319 257 y Fy(\))k(:=)h(\()p Fw(q)1450 240 y Fu(?)1479 257 y Fw(;)8 b Fv(\000)p Fw(q)r Fy(\))14 b(w)o(e)h(ma)o(y)f(use)i(the)-12 313 y(w)o(ell)g(kno)o(wn)f (form)o(ula)g(\(see)g([1)o(,)g(p.)g(124]\))649 407 y(gra)7 b Fw(')750 388 y Fu(\003)782 407 y Fy(=)13 b(\()p Fx(q)870 388 y Fu(?)902 407 y Fv(\002)d Fx(q)p Fy(\))g Fv(\011)h Fw(U)5 b Fy(\(gra)h Fw(')p Fy(\))i Fw(:)-12 509 y Fy(Therefore)18 b(\()p Fw(q)237 492 y Fu(?)266 509 y Fw(;)8 b(q)r Fy(\))16 b Fv(2)h Fy(gra)7 b Fw(')491 492 y Fu(\003)529 509 y Fy(i\013)586 462 y FB(D)611 509 y Fy(\()p Fw(q)651 492 y Fu(?)680 509 y Fw(;)h(q)r Fy(\))p Fw(;)g Fy(\()p Fw(Q)816 492 y Fu(?)843 509 y Fw(p;)g Fv(\000)p Fw(Qp)p Fy(\))999 462 y FB(E)1041 509 y Fy(=)17 b(0)h(for)f(all)i Fw(p)e Fv(2)h Fx(p)f Fy(i\013)i Fv(h)p Fw(q)1498 492 y Fu(?)1527 509 y Fw(;)8 b(p)p Fv(i)16 b Fy(=)h Fv(h)p Fw(q)r(;)8 b(p)p Fv(i)16 b Fy(for)i(all)-12 579 y Fw(p)12 b Fv(2)h Fx(p)i Fy(i\013)h Fw(p)182 562 y Fu(?)224 579 y Fy(:=)c Fw(q)306 562 y Fu(?)346 579 y Fv(\000)e Fw(q)15 b Fv(2)e Fx(p)492 562 y Fu(?)536 579 y Fy(i\013)i(\()p Fw(q)630 562 y Fu(?)659 579 y Fw(;)8 b(q)r Fy(\))k Fv(2)h Fy(gra)7 b Fw(\032)870 562 y Fu(\000)p Ft(1)916 579 y Fy(.)p 1857 579 24 24 v -12 712 a Fq(4.3)16 b(Rema)o(rk)23 b Fy(Note)15 b(that)g(the)g(preceding)i(lemma)f(dep)q(ends)h(only)f(on)f(the)g (assumption)h(that)e Fw(P)22 b Fy(and)16 b Fw(Q)f Fy(are)-12 769 y(in)i(generic)g(p)q(osition)g(\(the)f(di\013eren)o(t)g(role)g(of)g Fw(P)23 b Fy(and)16 b Fw(Q)g Fy(w.r.t.)e(\000,)i(i.e.)g Fw(Q)p Fy(\000)f(=)f(\000)p Fw(Q)j Fy(and)f(\000)p Fw(P)6 b Fy(\000)15 b(=)g Fw(P)1725 752 y Fu(?)1755 769 y Fy(,)g(is)i(not)-12 825 y(used\).)j(This)c(means)f(that)f(the)i(preceding)g(lemma)g (remains)f(true)h(if)f(w)o(e)g(do)g(the)h(follo)o(wing)f(replacemen)o (ts)647 919 y Fw(Q)e Fv(!)g Fw(P)97 b Fy(and)91 b Fw(P)20 b Fv(!)13 b Fw(Q)1187 900 y Fu(?)1224 919 y Fw(:)-12 1013 y Fy(Then)j(w)o(e)f(obtain:)-12 1111 y Fq(4.4)h(Co)o(rolla)o(ry)23 b Fs(Put)529 1173 y Fx(H)562 1180 y Fp(\025)597 1173 y Fy(:=)658 1126 y FB(n)685 1173 y Fy(\()p Fw(p;)8 b(p)770 1154 y Fu(?)798 1173 y Fy(\))13 b Fv(2)g Fx(p)c Fv(\002)i Fx(p)973 1154 y Fu(?)1015 1173 y Fv(j)h Fw(Q)1076 1154 y Fu(?)1106 1173 y Fw(p)g Fy(=)h Fw(Q)1225 1154 y Fu(?)1254 1173 y Fw(p)1277 1154 y Fu(?)1307 1126 y FB(o)1342 1173 y Fw(:)-12 1266 y Fs(Then)19 b Fx(H)143 1273 y Fp(\025)185 1266 y Fs(is)g(the)h(gr)n(aph)h(of)e(a)h(line)n(ar)f(inje)n(ctive)g (close)n(d)g(mapping)h Fw(\025)p Fy(:)26 b Fx(p)19 b Fv(!)g Fx(p)1377 1249 y Fu(?)1426 1266 y Fs(with)i(dense)e(domain)h (and)-12 1322 y(image.)h Fy(gra)7 b Fw(\025)16 b Fs(c)n(an)f(b)n(e)h(p) n(ar)n(ametrize)n(d)g(by)g Fx(q)p Fs(:)639 1421 y Fy(gra)6 b Fw(\025)13 b Fy(=)797 1374 y FB(n)824 1421 y Fy(\()p Fw(P)6 b(q)r(;)i Fv(\000)p Fw(P)990 1403 y Fu(?)1020 1421 y Fw(q)r Fy(\))k Fv(j)g Fw(q)j Fv(2)e Fx(q)1197 1374 y FB(o)1232 1421 y Fw(:)59 1533 y Fy(The)20 b(parametrization)g (of)f Fw(')h Fy(and)g Fw(\032)g Fy(in)h(terms)e(of)h Fx(p)g Fy(resp.)g Fx(p)1162 1516 y Fu(?)1211 1533 y Fy(suggests)f(to)h (consider)h(the)f(follo)o(wing)-12 1589 y(mappings:)791 1683 y Fw(Q)30 b Fy(:)41 b Fx(p)934 1664 y Fu(?)976 1683 y Fv(\000)-7 b(!)13 b Fx(q)731 b Fy(\(13\))791 1752 y Fw(Q)827 1736 y Fu(?)857 1752 y Fy(:)41 b Fx(p)934 1733 y Fu(?)976 1752 y Fv(\000)-7 b(!)13 b Fx(q)1084 1733 y Fu(?)1815 1752 y Fy(\(14\))791 1821 y Fw(Q)30 b Fy(:)41 b Fx(p)h Fv(\000)-7 b(!)13 b Fx(q)731 b Fy(\(15\))791 1890 y Fw(Q)827 1873 y Fu(?)857 1890 y Fy(:)41 b Fx(p)h Fv(\000)-7 b(!)13 b Fx(q)1084 1871 y Fu(?)1121 1890 y Fw(;)681 b Fy(\(16\))-12 1984 y(where)10 b(the)h(\014rst)e(t)o(w)o(o)g (are)h(related)h(to)e Fw(\032)h Fy(and)g(the)h(last)f(t)o(w)o(o)f(are)g (related)i(to)f Fw(')p Fy(.)17 b(F)l(or)10 b(example,)i(the)e (parametriza-)-12 2040 y(tion)15 b(of)f Fw(\032)h Fy(in)g(terms)f(of)h Fx(p)424 2024 y Fu(?)468 2040 y Fy(means)g(that)f Fw(\032)g Fy(can)h(b)q(e)g(seen)h(as)e(the)h(comp)q(osition)h(of)e(the)h(follo)o (wing)g(mappings:)649 2134 y Fw(QP)720 2116 y Fu(?)750 2134 y Fx(h)e Fv(\000)-7 b(!)13 b Fw(P)908 2116 y Fu(?)938 2134 y Fx(h)f Fv(\000)-7 b(!)13 b Fw(Q)1096 2116 y Fu(?)1125 2134 y Fw(P)1160 2116 y Fu(?)1190 2134 y Fx(h)8 b Fw(:)-12 2228 y Fy(Later)15 b(w)o(e)g(will)i(also)e(need)h(to)e(consider)j(the)e (mappings)821 2322 y Fw(P)6 b Fy(:)42 b Fx(q)933 2303 y Fu(?)975 2322 y Fv(\000)-7 b(!)13 b Fx(p)731 b Fy(\(17\))821 2391 y Fw(P)6 b Fy(:)42 b Fx(q)g Fv(\000)-7 b(!)13 b Fx(p)8 b Fw(:)710 b Fy(\(18\))59 2485 y(Due)18 b(to)h(the)g(fact)f (that)g Fx(q)h Fy(and)g Fx(p)g Fy(are)f(in)i(generic)g(p)q(osition)g (the)f(mappings)g(\(13\){\(18\))d(are)j(b)q(ounded,)-12 2542 y(injectiv)o(e)d(and)g(their)g(images)f(are)g(dense)h(in)g(the)f (corresp)q(onding)h(\014nal)g(subspaces.)59 2598 y(W)l(e)f(will)j(next) e(giv)o(e)g(a)f(criterion)i(for)e(the)h(bicon)o(tin)o(uit)o(y)h(of)e (the)h(mappings)g(\(13\){\(18\).)j(First)d(note)f(that)-12 2655 y(b)q(ecause)h Fw(P)22 b Fy(is)15 b(a)g(basis)h(pro)s(jection)449 2749 y Fv(k)p Fw(P)6 b(Q)p Fv(k)13 b Fy(=)g Fv(k)p Fw(QP)6 b Fv(k)12 b Fy(=)h Fv(k)p Fy(\(1)-11 b FF(l)15 b Fv(\000)c Fw(P)6 b Fy(\))p Fw(Q)p Fv(k)12 b Fy(=)h Fv(k)p Fw(Q)p Fy(\(1)-11 b FF(l)15 b Fv(\000)10 b Fw(P)c Fy(\))p Fv(k)13 b Fy(=:)g Fw(\016)-12 2843 y Fy(and)i(0)e Fw(<)g(\016)h Fv(\024)f Fy(1.)20 b(So)15 b(w)o(e)g(can)g(distinguish)i(b)q(et)o(w)o (een)f(the)f(t)o(w)o(o)f(cases:)20 b Fw(\016)14 b(<)f Fy(1)i(and)h Fw(\016)e Fy(=)f(1.)919 2967 y(10)p eop %%Page: 11 11 11 10 bop -12 50 a Fq(4.5)16 b(Prop)q(osition)24 b Fs(L)n(et)d Fw(P)q(;)8 b(Q)22 b Fs(and)f Fw(\016)j Fs(given)d(as)h(b)n(efor)n(e.)37 b(If)21 b Fw(\016)j(<)f Fy(1)p Fs(,)g(then)f(the)g(mappings)f (\(13\){\(18\))h(ar)n(e)-12 107 y(bic)n(ontinuous,)17 b(in)g(p)n(articular)h(their)f(images)g(c)n(oincide)g(with)h(the)f(c)n (orr)n(esp)n(onding)f(\014nal)g(sp)n(ac)n(es.)23 b(Mor)n(e)n(over,)-12 163 y(the)16 b(r)n(elations)456 220 y Fv(k)p Fw(P)h Fv(\000)10 b Fw(Q)p Fv(k)i Fy(=)h Fv(k)p Fy(\(1)-11 b FF(l)15 b Fv(\000)c Fw(Q)p Fy(\))p Fw(P)6 b Fv(k)12 b Fy(=)h Fv(k)p Fy(\(1)-11 b FF(l)15 b Fv(\000)c Fw(Q)p Fy(\)\(1)-12 b FF(l)16 b Fv(\000)10 b Fw(P)c Fy(\))p Fv(k)13 b Fy(=)g Fw(\016)-12 301 y Fs(hold.)-12 393 y(Pr)n(o)n(of:)18 b Fy(This)12 b(result)g(is)h(a)e(sp)q(ecial)i(case)f(of)f(Theorem)h (6.34)e(in)j([26)o(,)f(p.)f(56].)18 b(Note)12 b(that)f(the)g(second)h (alternativ)o(e)-12 449 y(stated)j(in)h(Kato's)e(result)i(cannot)f(app) q(ear)h(in)g(the)g(presen)o(t)f(situation,)g(as)g(a)g(consequence)i(of) e(the)h(fact)e(that)-12 506 y Fx(p)h Fy(and)g Fx(q)h Fy(are)e(in)j(generic)f(p)q(osition.)p 1857 506 24 24 v -12 644 a Fq(4.6)g(Rema)o(rk)23 b Fy(This)f(situation)f(corresp)q (onds)g(to)f(the)h(case)f(where)h(the)g(index)h(of)e Fw(P)28 b Fy(and)20 b Fw(Q)h Fy(is)g(0)g(\(cf.)f([6)o(,)-12 701 y(Theorem)15 b(3.3]\).)-12 804 y Fq(4.7)h(Prop)q(osition)24 b Fs(L)n(et)14 b Fw(P)q(;)8 b(Q)13 b Fs(and)h Fw(\016)i Fs(given)e(as)g(b)n(efor)n(e.)19 b(If)14 b Fw(\016)g Fy(=)f(1)p Fs(,)i(then)f(the)g(inverse)f(mappings)h(of)g(\(13\){\(18\)) -12 861 y(ar)n(e)i(unb)n(ounde)n(d)h(and)f(densely)f(de\014ne)n(d,)h (i.e.)g(the)h(images)f(of)g(\(13\){\(18\))g(ar)n(e)h(nontrivial)e(pr)n (op)n(er)i(dense)f(sets)-12 917 y(in)g(the)g(c)n(orr)n(esp)n(onding)f (\014nal)g(subsp)n(ac)n(es.)-12 1020 y(Pr)n(o)n(of:)30 b Fy(W)l(e)20 b(will)i(only)e(sho)o(w)g(the)g(assertion)g(for)f(the)i (mapping)f(\(18\),)g(since)h(one)g(can)f(easily)h(adapt)f(the)-12 1077 y(follo)o(wing)d(argumen)o(ts)e(to)h(the)g(other)g(cases.)22 b(Put)16 b Fw(A)f Fy(:=)f Fw(QP)1065 1060 y Fu(?)1095 1077 y Fw(Q)p Fh(\026)p Fx(q)g Fv(2)h(L)p Fy(\()p Fx(q)p Fy(\),)h(so)g(that)f Fw(A)f Fy(=)h Fw(A)1637 1060 y Fu(\003)1673 1077 y Fy(and)h Fw(A)f Fv(\025)f Fy(0.)-12 1133 y(F)l(rom)460 1190 y(spr)8 b Fw(A)13 b Fy(=)g Fv(k)p Fw(A)p Fv(k)f Fy(=)h Fv(k)p Fw(QP)858 1171 y Fu(?)887 1190 y Fw(P)922 1171 y Fu(?)952 1190 y Fw(Q)p Fv(k)g Fy(=)g Fv(k)p Fw(P)1130 1171 y Fu(?)1159 1190 y Fw(Q)p Fv(k)1218 1171 y Ft(2)1250 1190 y Fy(=)g Fw(\016)1320 1171 y Ft(2)1353 1190 y Fy(=)g(1)-12 1272 y(w)o(e)19 b(obtain)g(1)g Fv(2)h Fy(sp)q(ec)8 b Fw(A)p Fy(.)32 b(Ho)o(w)o(ev)o(er,)19 b(1)f(is)i(not)f(an)g(eigen)o(v)m (alue)i(of)e Fw(A)p Fy(,)g(b)q(ecause)i Fw(Aq)g Fy(=)e Fw(q)r Fy(,)h Fw(q)h Fv(2)f Fx(q)p Fy(,)g(implies)-12 1328 y(s)p Fv(\000)8 b Fy(lim)4 1357 y Fp(n)p Fu(!1)120 1328 y Fy(\()p Fw(QP)e Fy(\))227 1311 y Fp(n)250 1328 y Fw(q)19 b Fy(=)e Fw(q)i Fy(and)f(this)g(means)g Fw(q)h Fv(2)e Fx(q)12 b Fv(\\)g Fx(p)889 1311 y Fu(?)935 1328 y Fy(=)17 b Fv(f)p Fy(0)p Fv(g)p Fy(.)27 b(Th)o(us)18 b(k)o(er)7 b(\(1)-11 b FF(l)1329 1335 y Fo(q)1360 1328 y Fv(\000)12 b Fw(A)p Fy(\))k(=)i Fv(f)p Fy(0)p Fv(g)f Fy(or)g(\(1)-11 b FF(l)1719 1335 y Fo(q)1750 1328 y Fv(\000)12 b Fw(A)p Fy(\))1849 1311 y Fu(\000)p Ft(1)-12 1403 y Fy(exists)i(and)g(is)h(un)o(b)q(ounded)g(since)g(1)j Fw(=)-28 b Fv(2)13 b Fy(res)7 b Fw(A)p Fy(.)20 b(Therefore)14 b Fw(#)e Fy(:=)h(dom)7 b(\(1)-11 b FF(l)1241 1410 y Fo(q)1268 1403 y Fv(\000)8 b Fw(A)p Fy(\))1363 1386 y Fu(\000)p Ft(1)1424 1403 y Fy(is)14 b(a)g(prop)q(er)g(dense)g(subset)-12 1459 y(in)h Fx(q)g Fy(and)f(this)h(means)f(ima)8 b(\(1)-11 b FF(l)518 1466 y Fo(q)545 1459 y Fv(\000)9 b Fw(A)p Fy(\))j(=)h Fw(#)g Fy(=)g(ima)8 b(\()p Fw(Q)g Fv(\000)g Fw(QP)1046 1443 y Fu(?)1076 1459 y Fw(Q)p Fy(\))13 b(=)g(ima)7 b(\()p Fw(QP)f(Q)p Fy(\).)20 b(Finally)l(,)15 b(from)f(the)g(p)q(olar) -12 1516 y(decomp)q(osition)i(of)f Fw(P)6 b(Q)p Fy(,)674 1577 y Fw(P)g(Q)13 b Fy(=)g(sgn)8 b(\()p Fw(P)e(Q)p Fy(\))k Fv(\001)f Fy(\()p Fw(QP)d(Q)p Fy(\))1167 1545 y Fk(1)p 1167 1551 16 2 v 1167 1571 a(2)1197 1577 y Fw(;)-12 1666 y Fy(w)o(e)20 b(ha)o(v)o(e)g(that)g(sgn)8 b(\()p Fw(P)e(Q)p Fy(\))20 b(maps)g(ima)8 b(\()p Fw(QP)e(Q)p Fy(\))828 1634 y Fk(1)p 828 1640 V 828 1661 a(2)871 1666 y Fy(isometrically)22 b(on)o(to)d(ima)8 b(\()p Fw(P)e(Q)p Fy(\))22 b(=)g Fw(P)6 b Fx(q)p Fy(.)36 b(Th)o(us)21 b Fw(P)6 b Fx(q)21 b Fy(is)g(a)-12 1722 y(prop)q(er)15 b(dense)h(set)f(in)h Fx(p)p Fy(,)f(i.e.)g Fw(P)6 b Fy(:)i Fx(q)13 b Fv(!)g Fx(p)i Fy(is)g(un)o(b)q(ounded)i(in)o (v)o(ertible.)p 1857 1722 24 24 v -12 1861 a Fq(4.8)f(Rema)o(rk)23 b Fy(\(i\))e(Note)g(that)f(if)h(dim)8 b Fx(h)22 b Fw(<)h Fv(1)p Fy(,)f(then)f(the)g(case)g Fv(k)p Fw(P)6 b(Q)p Fv(k)22 b Fy(=)g(1)f(is)g(not)g(p)q(ossible,)j(since)e(the)-12 1917 y(corresp)q(onding)17 b(op)q(erators)e(can)i(not)f(ha)o(v)o(e)f (con)o(tin)o(uous)i(sp)q(ectrum.)23 b(It)17 b(is)f(easy)g(to)g(sho)o(w) f(that)h(in)h(this)g(case)-12 1974 y Fx(p)10 b Fv(\\)g Fx(q)j Fy(=)g Fv(f)p Fy(0)p Fv(g)h Fy(i\013)i Fv(k)p Fw(P)6 b(Q)p Fv(k)12 b Fw(<)h Fy(1.)59 2030 y(\(ii\))j(Note)g(also)h (that)f Fv(k)p Fy(\(1)-12 b FF(l)16 b Fv(\000)c Fw(P)6 b Fy(\))p Fw(Q)p Fv(k)14 b Fy(=)h(1)h(implies)j Fv(k)p Fy(\(1)-12 b FF(l)16 b Fv(\000)c Fw(Q)p Fy(\))p Fw(P)6 b Fv(k)14 b Fy(=)h(1,)h(b)q(ecause)h(otherwise)g(b)o(y)f(Prop)q(osi-) -12 2087 y(tion)f(4.5)g Fv(k)p Fy(\(1)-12 b FF(l)16 b Fv(\000)10 b Fw(Q)p Fy(\))p Fw(P)c Fv(k)13 b Fw(<)g Fy(1)h(implies)k Fv(k)p Fy(\(1)-11 b FF(l)15 b Fv(\000)10 b Fw(P)c Fy(\))p Fw(Q)p Fv(k)13 b Fw(<)g Fy(1.)-12 2190 y Fq(4.9)j(Prop)q(osition)24 b Fs(L)n(et)i(the)g(pr)n(oje)n(ctions)g Fw(P)q(;)8 b(Q)26 b Fs(and)g(the)h(mappings)f Fw(';)8 b(\032)25 b Fs(b)n(e)h(given)g(as)g (b)n(efor)n(e.)50 b(Then)-12 2246 y Fw(';)8 b(\032)p Fy(:)22 b Fx(q)13 b Fv(!)g Fx(q)213 2230 y Fu(?)259 2246 y Fs(ar)n(e)j(bic)n(ontinuous)g(i\013)g Fv(k)p Fw(P)6 b(Q)p Fv(k)12 b Fw(<)h Fy(1)p Fs(.)-12 2350 y(Pr)n(o)n(of:)19 b Fy(Supp)q(ose)d(that)e Fv(k)p Fw(P)6 b(Q)p Fv(k)12 b Fw(<)h Fy(1,)h(so)g(that)g(b)o(y)g(Prop)q(osition)h(4.5)f(w)o(e)g(ha) o(v)o(e)g(that)g(the)g(mappings)h(\(13\){\(18\))-12 2406 y(are)c(bicon)o(tin)o(uous.)20 b(But)11 b(as)h(men)o(tioned)g(b)q (efore)g(w)o(e)f(kno)o(w)g(that)f Fw(')i Fy(as)f(w)o(ell)h(as)f Fw(\032)g Fy(can)h(b)q(e)g(seen)g(as)f(comp)q(osition)-12 2462 y(of)k(the)g(mappings)628 2562 y Fw(')p Fy(:)41 b Fw(QP)6 b Fx(h)42 b Fv(\000)-7 b(!)13 b Fw(P)6 b Fx(h)43 b Fv(\000)-8 b(!)13 b Fw(Q)1158 2543 y Fu(?)1188 2562 y Fw(P)6 b Fx(h)631 2630 y Fw(\032)p Fy(:)44 b Fw(QP)783 2612 y Fu(?)813 2630 y Fx(h)12 b Fv(\000)-7 b(!)13 b Fw(P)970 2612 y Fu(?)1000 2630 y Fx(h)g Fv(\000)-8 b(!)13 b Fw(Q)1158 2612 y Fu(?)1188 2630 y Fw(P)1223 2612 y Fu(?)1253 2630 y Fx(h)8 b Fw(;)-12 2730 y Fy(hence)16 b(they)f(m)o(ust)g(b)q(e)h(bicon)o(tin)o(uous.)59 2786 y(In)f(the)f(case)g(that)g Fw(\032)g Fy(and)h Fw(')f Fy(are)g(bicon)o(tin)o(uous,)h(then)g(dom)7 b Fw(\032)12 b Fy(=)h(dom)8 b Fw(')k Fy(=)h Fx(q)i Fy(and)f(ima)8 b Fw(\032)k Fy(=)h(ima)8 b Fw(')k Fy(=)h Fx(q)1853 2770 y Fu(?)1883 2786 y Fy(.)-12 2843 y(Finally)l(,)j(Prop)q(osition)g(4.7)e (implies)k Fv(k)p Fw(P)6 b(Q)p Fv(k)12 b Fw(<)h Fy(1.)p 1857 2843 V 919 2967 a(11)p eop %%Page: 12 12 12 11 bop 59 50 a Fy(Motiv)m(ated)11 b(b)o(y)g(Lemma)g(3.5)f(w)o(e)h (will)h(analyze)g(next)f(the)g(an)o(tilinear)i(mappings)e(de\014ned)i (b)o(y)e(the)g(follo)o(wing)-12 107 y(graphs:)466 207 y(gra)c Fw(\014)44 b Fy(:=)696 160 y FB(n)724 207 y Fy(\()p Fw(P)6 b(q)r(;)i(P)e Fy(\000)p Fw(q)r Fy(\))12 b Fv(2)h Fx(p)d Fv(\002)h Fx(p)h Fv(j)g Fw(q)j Fv(2)e Fx(q)1217 160 y FB(o)465 298 y Fy(gra)7 b Fw(\013)42 b Fy(:=)696 251 y FB(n)724 298 y Fy(\()p Fw(P)6 b(q)799 279 y Fu(?)829 298 y Fw(;)i Fv(\000)p Fw(P)e Fy(\000)p Fw(q)970 279 y Fu(?)1000 298 y Fy(\))12 b Fv(2)h Fx(p)d Fv(\002)g Fx(p)i Fv(j)h Fw(q)1234 279 y Fu(?)1276 298 y Fv(2)g Fx(q)1341 279 y Fu(?)1370 251 y FB(o)1406 298 y Fw(:)-12 401 y Fy(\(Note)20 b(that)g(the)h(r.h.s.)e(of)i(the)f(preceding)j (equations)e(de\014ne)g(indeed)i(graphs)d(of)h(an)o(tilinear)g (mappings,)-12 457 y(b)q(ecause)16 b(the)f(assignmen)o(ts)g Fw(q)g Fv(!)e Fw(P)6 b(q)18 b Fy(and)d Fw(q)762 441 y Fu(?)804 457 y Fv(!)e Fw(P)6 b(q)919 441 y Fu(?)964 457 y Fy(are)15 b(injectiv)o(e.\))-12 556 y Fq(4.10)h(Lemma)24 b Fs(The)c(mappings)g Fw(\013;)8 b(\014)21 b Fs(de\014ne)n(d)e(by)h (the)g(pr)n(e)n(c)n(e)n(ding)e(gr)n(aphs)i(ar)n(e)f(anti{line)n(ar,)h (inje)n(ctive)f(and)-12 613 y(close)n(d)d(with)i(dense)e(domains)h(and) g(images)f Fy(dom)8 b Fw(\013)14 b Fy(=)g(ima)8 b Fw(\013)14 b Fy(=)g Fw(P)6 b Fy(\()p Fx(q)1203 596 y Fu(?)1233 613 y Fy(\))p Fs(,)17 b Fy(dom)8 b Fw(\014)16 b Fy(=)e(ima)8 b Fw(\014)16 b Fy(=)e Fw(P)6 b Fx(q)p Fs(.)24 b(F)m(urther,)-12 669 y(we)16 b(have)h Fw(\013)189 653 y Ft(2)221 669 y Fy(=)c(id)q Fs(,)k Fw(\014)366 653 y Ft(2)398 669 y Fy(=)c(id)k Fs(on)f Fw(P)6 b Fy(\()p Fx(q)640 653 y Fu(?)670 669 y Fy(\))16 b Fs(r)n(esp.)g Fw(P)6 b Fx(q)17 b Fs(and)f Fw(\013)d Fy(=)g Fw(\014)1094 653 y Fu(\003)1114 669 y Fs(.)-12 768 y(Pr)n(o)n(of:)27 b Fy(W)l(e)19 b(will)h(only)f(pro)o(v) o(e)f(the)h(last)f(equation,)i(b)q(ecause)g(the)e(other)h(statemen)o (ts)e(follo)o(w)i(immediately)-12 825 y(from)e(the)g(de\014nition.)28 b(No)o(w)17 b(b)o(y)g(de\014nition)i(w)o(e)e(ha)o(v)o(e)g(\()p Fw(p)996 832 y Ft(0)1016 825 y Fw(;)8 b(p)1060 832 y Ft(1)1078 825 y Fy(\))16 b Fv(2)h Fy(gra)7 b Fw(\014)1258 808 y Fu(\003)1295 825 y Fy(i\013)17 b Fv(h)p Fw(p)1392 832 y Ft(0)1412 825 y Fw(;)8 b(P)e Fy(\000)p Fw(q)r Fv(i)16 b Fy(=)g Fv(h)p Fw(P)6 b(q)r(;)i(p)1722 832 y Ft(1)1741 825 y Fv(i)17 b Fy(for)g(all)-12 881 y Fw(q)d Fv(2)f Fx(q)j Fy(i\013)f Fw(q)179 865 y Fu(?)221 881 y Fy(:=)e Fw(p)305 888 y Ft(0)334 881 y Fv(\000)e Fy(\000)p Fw(p)431 888 y Ft(1)464 881 y Fv(2)h Fx(q)528 865 y Fu(?)573 881 y Fy(i\013)k(\()p Fw(p)669 888 y Ft(0)688 881 y Fw(;)8 b(p)732 888 y Ft(1)751 881 y Fy(\))k Fv(2)h Fy(gra)7 b Fw(\013)p Fy(.)p 1857 881 24 24 v -12 1016 a Fq(4.11)16 b(Rema)o(rk)23 b Fy(\(i\))h(Recall)h(Subsection)g(3.1)e(and)h(denote)f (b)o(y)h Fw(S)i Fy(the)e(T)l(omita)f(op)q(erator)g(asso)q(ciated)g(to) -12 1072 y(\()p Fv(M)p Fy(\()p Fx(q)p Fy(\))p Fw(;)8 b Fy(\012\).)17 b(Then)f(from)e(the)i(preceding)g(result)g(w)o(e)f(ha)o (v)o(e)g Fw(S)s Fh(\026)p Fx(p)d Fv(\023)h Fw(\014)k Fy(and)e Fw(S)1324 1056 y Fu(\003)1343 1072 y Fh(\026)p Fx(p)e Fv(\023)g Fw(\013)p Fy(.)59 1129 y(\(ii\))20 b(Using)h(the)f (mappings)h(\(17\))e(and)h(\(18\))f(w)o(e)h(can)g(no)o(w)f(state)h (similarly)i(as)d(in)i(Prop)q(osition)g(4.9)e(a)-12 1185 y(criterion)d(for)f(the)g(bicon)o(tin)o(uit)o(y)h(of)f Fw(\013;)8 b(\014)r Fy(:)20 b(the)15 b(mappings)h Fw(\013;)8 b(\014)16 b Fy(are)f(bicon)o(tin)o(uous)i(i\013)e Fv(k)p Fw(P)6 b(Q)p Fv(k)12 b Fw(<)h Fy(1.)-12 1284 y Fq(4.12)j(Prop)q (osition)24 b Fs(The)16 b(mapping)f Fw(\014)656 1268 y Fu(\003)669 1284 y Fw(\014)r Fy(:)23 b Fx(p)12 b Fv(!)h Fx(p)i Fs(is)g(a)h(densely)f(de\014ne)n(d)f(line)n(ar)h(p)n(ositive)g (self{adjoint)g(op)n(er-)-12 1341 y(ator)i(on)f Fx(p)g Fs(with)h(gr)n(aph)598 1402 y Fy(gra)7 b Fw(\014)697 1384 y Fu(\003)709 1402 y Fw(\014)15 b Fy(=)798 1355 y FB(n)825 1402 y Fy(\()p Fw(P)6 b(Qp;)i(P)e(Q)1029 1384 y Fu(?)1058 1402 y Fw(p)p Fy(\))13 b Fv(j)f Fw(p)g Fv(2)h Fx(p)1238 1355 y FB(o)1273 1402 y Fw(:)-12 1494 y Fs(Mor)n(e)n(over,)j Fy(\()p Fw(\014)248 1477 y Fu(\003)260 1494 y Fw(\014)r Fy(\))306 1477 y Fu(\000)p Ft(1)365 1494 y Fy(=)d Fw(\014)r(\014)469 1477 y Fu(\003)502 1494 y Fy(=)g Fw(\013)579 1477 y Fu(\003)591 1494 y Fw(\013)p Fs(.)-12 1593 y(Pr)n(o)n(of:)20 b Fy(W)l(e)15 b(will)i(compute)e(\014rst)g(the)g(domain)h(of)f Fw(\014)890 1576 y Fu(\003)902 1593 y Fw(\014)r Fy(.)20 b(Recalling)e(that)c Fw(\014)1287 1576 y Fu(\003)1320 1593 y Fy(=)f Fw(\013)i Fy(w)o(e)g(ha)o(v)o(e)317 1693 y(dom)7 b(\()p Fw(\014)456 1674 y Fu(\003)468 1693 y Fw(\014)r Fy(\))41 b(=)632 1646 y FB(n)660 1693 y Fw(P)6 b(q)15 b Fv(j)d Fw(q)j Fv(2)e Fx(q)g Fy(and)f Fw(P)6 b Fy(\000)p Fw(q)16 b Fv(2)d Fy(dom)7 b Fw(\013)13 b Fy(=)g Fw(P)6 b Fy(\()p Fx(q)1353 1674 y Fu(?)1383 1693 y Fy(\))1401 1646 y FB(o)555 1808 y Fy(=)632 1761 y FB(n)660 1808 y Fw(P)g(q)15 b Fv(j)d Fw(q)j Fv(2)e Fx(q)g Fy(and)f Fw(P)6 b Fy(\000)p Fw(q)16 b Fy(=)d Fw(P)6 b(q)1157 1789 y Fu(?)1199 1808 y Fy(for)15 b(some)d Fw(q)1402 1789 y Fu(?)1444 1808 y Fv(2)h Fx(q)1509 1789 y Fu(?)1539 1761 y FB(o)555 1923 y Fy(=)632 1876 y FB(n)660 1923 y Fw(P)6 b(q)15 b Fv(j)d Fw(q)j Fv(2)e Fx(q)g Fy(and)f(\000)p Fw(q)j Fv(2)e Fy(dom)8 b Fw(\032)k Fy(=)h Fw(Q)p Fy(\()p Fx(p)1314 1904 y Fu(?)1343 1923 y Fy(\))1361 1876 y FB(o)555 2038 y Fy(=)632 1991 y FB(n)660 2038 y Fw(P)6 b(q)15 b Fv(j)d Fw(q)j Fv(2)e Fx(q)g Fy(and)f Fw(q)j Fv(2)e Fy(\000)p Fw(Q)p Fy(\()p Fx(p)1136 2019 y Fu(?)1165 2038 y Fy(\))g(=)g Fw(Q)p Fy(\(\000)p Fx(p)1349 2019 y Fu(?)1378 2038 y Fy(\))f(=)h Fw(Q)p Fx(p)1515 1991 y FB(o)555 2135 y Fy(=)42 b Fw(P)6 b(Q)p Fx(p)49 b Fy(=)f Fw(P)6 b(QP)g Fx(h)i Fw(;)-12 2230 y Fy(whic)o(h)18 b(is)g(dense)g(in)g Fx(p)p Fy(.)26 b(F)l(urthermore,)17 b(since)i Fw(P)6 b Fy(\000)p Fw(Qp)17 b Fy(=)g Fv(\000)p Fw(P)6 b(Q)1101 2214 y Fu(?)1131 2230 y Fy(\000)p Fw(p)p Fy(,)18 b Fw(p)e Fv(2)g Fx(p)h Fy(\(recall)i Fw(P)6 b Fy(\000)p Fw(p)17 b Fy(=)f(0,)h Fw(p)f Fv(2)h Fx(p)p Fy(\),)g(w)o(e)-12 2287 y(ha)o(v)o(e)213 2296 y FB(\020)238 2343 y Fw(\014)266 2325 y Fu(\003)278 2343 y Fw(\014)306 2296 y FB(\021)331 2343 y Fy(\()p Fw(P)6 b(Qp)p Fy(\))13 b(=)g Fw(\013)551 2296 y FB(\020)576 2343 y Fw(P)6 b Fy(\000)p Fw(Qp)698 2296 y FB(\021)736 2343 y Fy(=)13 b Fv(\000)p Fw(\013)848 2296 y FB(\020)873 2343 y Fw(P)6 b(Q)944 2325 y Fu(?)974 2343 y Fy(\000)p Fw(p)1025 2296 y FB(\021)1063 2343 y Fy(=)13 b Fw(P)6 b Fy(\000)p Fw(Q)1210 2325 y Fu(?)1240 2343 y Fy(\000)p Fw(p)13 b Fy(=)g Fw(P)6 b(Q)1423 2325 y Fu(?)1453 2343 y Fw(p)i(;)52 b(p)12 b Fv(2)h Fx(p)8 b Fw(:)-12 2429 y Fy(The)13 b(last)g(equations)g (concerning)h(the)e(in)o(v)o(erse)i(of)e Fw(\014)890 2412 y Fu(\003)902 2429 y Fw(\014)j Fy(follo)o(w)e(from)f(the)h (preceding)h(computation)e(and)h(from)-12 2485 y(the)i(fact)g(that)f Fw(\013)283 2469 y Ft(2)316 2485 y Fy(=)f(id)j(and)f Fw(\014)533 2469 y Ft(2)566 2485 y Fy(=)e(id)j(on)f(the)g(corresp)q (onding)h(domains)g(\(recall)g(Lemma)f(4.10\).)p 1857 2485 V 59 2577 a(In)g(the)h(follo)o(wing)g(w)o(e)e(will)j(use)f(the)f (notation)832 2672 y(\001)870 2679 y Fo(p)902 2672 y Fy(:=)e Fw(\014)991 2653 y Fu(\003)1003 2672 y Fw(\014)d(;)-12 2786 y Fy(where)15 b(dom)8 b(\001)251 2793 y Fo(p)283 2786 y Fy(=)13 b Fw(P)6 b(Q)p Fx(p)15 b Fy(is)h(dense)g(in)g Fx(p)p Fy(.)j(Note)c(further)g(that)g(\001)1118 2757 y Fu(\000)1150 2744 y Fk(1)p 1150 2750 16 2 v 1150 2770 a(2)1118 2794 y Fo(p)1185 2786 y Fy(=)e Fv(j)p Fw(\013)p Fv(j)h Fy(and)i(th)o(us)f(dom)7 b(\001)1622 2757 y Fu(\000)1654 2744 y Fk(1)p 1655 2750 V 1655 2770 a(2)1622 2794 y Fo(p)1689 2786 y Fy(=)13 b Fw(P)6 b Fy(\()p Fx(q)1812 2770 y Fu(?)1842 2786 y Fy(\).)59 2843 y(Next)15 b(w)o(e)g(will)h(calculate)h(the)e (graph)g(of)g(the)g(p)q(ositiv)o(e)h(self{adjoin)o(t)g(op)q(erator)e Fw(')1472 2826 y Fu(\003)1491 2843 y Fw(')p Fy(:)8 b Fx(q)k Fv(!)h Fx(q)p Fy(.)919 2967 y(12)p eop %%Page: 13 13 13 12 bop -12 50 a Fq(4.13)16 b(Prop)q(osition)24 b Fs(The)16 b(gr)n(aph)h(of)f Fw(')653 34 y Fu(\003)673 50 y Fw(')g Fs(is)g(given)f(by)598 158 y Fy(gra)7 b Fw(')699 139 y Fu(\003)711 158 y Fw(')12 b Fy(=)801 111 y FB(n)829 158 y Fy(\()p Fw(QP)6 b(q)r(;)i(QP)1032 139 y Fu(?)1061 158 y Fw(q)r Fy(\))k Fv(j)g Fw(q)j Fv(2)e Fx(q)1238 111 y FB(o)1273 158 y Fw(:)-12 271 y Fs(Pr)n(o)n(of:)20 b Fy(W)l(e)15 b(b)q(egin)i(computing)e(dom)8 b(\()p Fw(')695 255 y Fu(\003)706 271 y Fw(')p Fy(\).)20 b(Since)c(b)o(y)f(Lemma)h(4.2) e Fw(')1234 255 y Fu(\003)1266 271 y Fy(=)f Fw(\032)1338 255 y Fu(\000)p Ft(1)1400 271 y Fy(w)o(e)i(ha)o(v)o(e)296 381 y(dom)8 b(\()p Fw(')438 362 y Fu(\003)449 381 y Fw(')p Fy(\))41 b(=)615 334 y FB(n)643 381 y Fw(Qp)12 b Fv(j)g Fw(p)h Fv(2)g Fx(p)f Fy(and)h Fw(Q)975 362 y Fu(?)1005 381 y Fw(p)f Fv(2)h Fy(dom)7 b Fw(\032)1200 362 y Fu(\000)p Ft(1)1260 381 y Fy(=)13 b Fw(Q)1344 362 y Fu(?)1373 381 y Fx(p)1396 362 y Fu(?)1425 334 y FB(o)538 496 y Fy(=)615 449 y FB(n)643 496 y Fw(Qp)f Fv(j)g Fw(p)h Fv(2)g Fx(p)f Fy(and)h Fw(Q)975 477 y Fu(?)1005 496 y Fw(p)f Fy(=)h Fw(Q)1124 477 y Fu(?)1153 496 y Fw(p)1176 477 y Fu(?)1219 496 y Fy(for)h(some)e Fw(p)1422 477 y Fu(?)1464 496 y Fv(2)h Fx(p)1530 477 y Fu(?)1560 449 y FB(o)538 611 y Fy(=)615 564 y FB(n)643 611 y Fw(Qp)f Fv(j)g Fw(p)h Fy(=)g Fw(P)6 b(q)15 b Fy(for)g(some)d Fw(q)i Fv(2)f Fx(q)1173 564 y FB(o)538 709 y Fy(=)42 b Fw(QP)6 b Fx(q)49 b Fy(=)f Fw(QP)6 b(Q)p Fx(h)i Fw(;)-12 811 y Fy(where)21 b(for)g(the)g(third)h (equation)f(w)o(e)g(ha)o(v)o(e)g(used)g(Corollary)g(4.4.)37 b(Using)22 b(again)f(this)g(corollary)h(w)o(e)f(can)-12 867 y(calculate)267 924 y Fw(\032)291 905 y Fu(\000)p Ft(1)338 877 y FB(\020)363 924 y Fw(')p Fy(\()p Fw(QP)6 b(q)r Fy(\))522 877 y FB(\021)559 924 y Fy(=)13 b Fw(\032)631 905 y Fu(\000)p Ft(1)677 924 y Fy(\()p Fw(Q)731 905 y Fu(?)761 924 y Fw(P)6 b(q)r Fy(\))12 b(=)h Fw(\032)920 905 y Fu(\000)p Ft(1)967 877 y FB(\020)992 924 y Fw(Q)1028 905 y Fu(?)1057 924 y Fy(\()p Fv(\000)p Fw(P)1145 905 y Fu(?)1175 924 y Fw(q)r Fy(\))1215 877 y FB(\021)1252 924 y Fy(=)g Fw(QP)1371 905 y Fu(?)1401 924 y Fw(q)d(;)52 b(q)15 b Fv(2)e Fx(q)8 b Fw(;)-12 1012 y Fy(and)15 b(the)h(pro)q(of)e (is)i(concluded.)p 1857 1012 24 24 v 59 1104 a(No)o(w)d(w)o(e)h(can)g (relate)g Fw(')g Fy(and)g Fw(')595 1088 y Fu(\003)614 1104 y Fw(')g Fy(with)g(\001)798 1111 y Fo(p)832 1104 y Fy(just)f(computing)i(the)f(orthogonal)f(decomp)q(osition)i(of)f Fw(')p Fy(\()p Fw(q)r Fy(\),)-12 1161 y Fw(q)g Fv(2)f Fy(dom)8 b Fw(')p Fy(,)14 b(resp.)h Fw(')355 1144 y Fu(\003)375 1161 y Fw(')p Fy(\()p Fw(q)r Fy(\),)f Fw(q)g Fv(2)f Fy(dom)7 b Fw(')690 1144 y Fu(\003)710 1161 y Fw(')p Fy(,)14 b(w.r.t.)g Fx(h)e Fy(=)h Fx(p)d Fv(\010)g Fx(p)1074 1144 y Fu(?)1104 1161 y Fy(.)-12 1255 y Fq(4.14)16 b(Co)o(rolla)o(ry)23 b Fs(Using)15 b(the)i(notation)f(b)n(efor)n(e)f(we)i(have)f(the)h(fol)r (lowing)f(formulas:)495 1357 y Fw(')p Fy(\()p Fw(Qp)p Fy(\))41 b(=)g(\001)775 1364 y Fo(p)795 1357 y Fy(\()p Fw(P)6 b(Qp)p Fy(\))j Fv(\000)i Fw(P)1015 1338 y Fu(?)1045 1357 y Fw(Qp)d(;)53 b(p)12 b Fv(2)h Fx(p)8 b Fw(;)515 b Fy(\(19\))362 1412 y FB(\020)386 1459 y Fw(')416 1440 y Fu(\003)436 1459 y Fw(')466 1412 y FB(\021)490 1459 y Fy(\()p Fw(QP)6 b(q)r Fy(\))42 b(=)f(\001)775 1466 y Fo(p)795 1459 y Fy(\()p Fw(P)6 b(QP)g(q)r Fy(\))k(+)g(\000)e(\001) 1088 1440 y Fu(\000)p Ft(1)1088 1470 y Fo(p)1135 1459 y Fy(\()p Fw(P)e Fy(\000)p Fw(QP)g(q)r Fy(\))i Fw(;)54 b(q)15 b Fv(2)e Fx(q)8 b Fw(:)-12 1566 y Fs(Pr)n(o)n(of:)20 b Fy(F)l(rom)14 b(Lemma)i(4.2)e(as)h(w)o(ell)h(as)f(Prop)q(osition)g (4.12)f(w)o(e)h(ha)o(v)o(e)298 1668 y Fw(')p Fy(\()p Fw(Qp)p Fy(\))d(=)h Fw(Q)519 1650 y Fu(?)548 1668 y Fw(p)f Fy(=)h Fw(P)6 b(Q)702 1650 y Fu(?)732 1668 y Fw(p)k Fy(+)h Fw(P)846 1650 y Fu(?)876 1668 y Fw(Q)912 1650 y Fu(?)941 1668 y Fw(p)i Fy(=)g(\001)1063 1675 y Fo(p)1082 1668 y Fy(\()p Fw(P)6 b(Qp)p Fy(\))k Fv(\000)g Fw(P)1302 1650 y Fu(?)1332 1668 y Fw(Qp)e(;)52 b(p)13 b Fv(2)g Fx(p)8 b Fw(:)-12 1770 y Fy(F)l(urther,)15 b(from)f(the)h(preceding)i(prop)q (osition)f(w)o(e)f(also)g(ha)o(v)o(e)g(for)f(an)o(y)h Fw(q)g Fv(2)e Fx(q)426 1831 y FB(\020)451 1878 y Fw(')481 1859 y Fu(\003)500 1878 y Fw(')530 1831 y FB(\021)555 1878 y Fy(\()p Fw(QP)6 b(q)r Fy(\))41 b(=)h Fw(QP)873 1859 y Fu(?)903 1878 y Fw(q)50 b Fy(=)e Fw(P)6 b(QP)1162 1859 y Fu(?)1192 1878 y Fw(q)12 b Fy(+)f Fw(P)1305 1859 y Fu(?)1335 1878 y Fw(QP)1406 1859 y Fu(?)1436 1878 y Fw(q)725 1962 y Fy(=)42 b Fw(P)6 b(Q)873 1943 y Fu(?)903 1962 y Fw(P)g(q)12 b Fy(+)f(\000)p Fw(P)6 b(QP)g Fy(\000)p Fw(q)725 2031 y Fy(=)42 b(\001)840 2038 y Fo(p)859 2031 y Fy(\()p Fw(P)6 b(QP)g(q)r Fy(\))k(+)h(\000)d(\001)1153 2012 y Fu(\000)p Ft(1)1153 2042 y Fo(p)1200 2031 y Fy(\()p Fw(P)e(Q)1289 2012 y Fu(?)1318 2031 y Fw(P)g Fy(\000)p Fw(q)r Fy(\))725 2107 y(=)42 b(\001)840 2114 y Fo(p)859 2107 y Fy(\()p Fw(P)6 b(QP)g(q)r Fy(\))k(+)h(\000)d(\001)1153 2088 y Fu(\000)p Ft(1)1153 2118 y Fo(p)1200 2107 y Fy(\()p Fw(P)e(QP)1324 2088 y Fu(?)1354 2107 y Fy(\000)p Fw(q)r Fy(\))725 2181 y(=)42 b(\001)840 2188 y Fo(p)859 2181 y Fy(\()p Fw(P)6 b(QP)g(q)r Fy(\))k(+)h(\000)d(\001)1153 2163 y Fu(\000)p Ft(1)1153 2193 y Fo(p)1200 2181 y Fy(\()p Fw(P)e Fy(\000)p Fw(QP)g(q)r Fy(\))i Fw(;)-12 2284 y Fy(whic)o(h)16 b(pro)o(v)o(es)f(the)g(second)h(form)o(ula.)p 1857 2284 V 59 2390 a(W)l(e)d(will)h(giv)o(e)g(next)f(t)o(w)o(o)e(form) o(ulas)i(in)h(terms)e(of)h(\001)944 2348 y Fk(1)p 944 2354 16 2 v 944 2375 a(2)939 2399 y Fo(p)979 2390 y Fy(for)f(the)h (comp)q(onen)o(ts)g(of)g(the)g(p)q(olar)g(decomp)q(osition)-12 2457 y(of)g Fw(')p Fy(.)19 b(Denote)13 b Fw(')g Fy(=)g(sgn)7 b Fw(')g Fv(\001)g(j)p Fw(')p Fv(j)p Fy(,)12 b(where)i(as)f(usual)h Fv(j)p Fw(')p Fv(j)e Fy(:=)h(\()p Fw(')1034 2440 y Fu(\003)1053 2457 y Fw(')p Fy(\))1106 2425 y Fk(1)p 1105 2431 V 1105 2452 a(2)1127 2457 y Fy(.)20 b(Recall)15 b(from)e(the)h(results)g(in)g (this)g(section)-12 2513 y(that)405 2615 y(dom)7 b Fw(')36 b Fy(=)h(dom)7 b Fv(j)p Fw(')p Fv(j)36 b Fy(=)g Fw(Q)p Fx(p)41 b Fv(\023)h Fw(QP)6 b Fx(q)37 b Fy(=)g(dom)7 b(\()p Fw(')1412 2596 y Fu(\003)1431 2615 y Fw(')p Fy(\))409 2708 y(dom)h Fw(\014)38 b Fy(=)f(dom)7 b Fv(j)p Fw(\014)r Fv(j)36 b Fy(=)g Fw(P)6 b Fx(q)42 b Fv(\023)g Fw(P)6 b(Q)p Fx(p)37 b Fy(=)f(dom)8 b(\001)1403 2715 y Fo(p)1430 2708 y Fw(:)919 2967 y Fy(13)p eop %%Page: 14 14 14 13 bop -12 50 a Fq(4.15)16 b(Theo)o(rem)23 b Fs(With)17 b(the)g(notation)f(ab)n(ove)g(we)h(have:)366 150 y Fv(j)p Fw(')p Fv(j)p Fy(\()p Fw(q)r Fy(\))40 b(=)h(\001)639 108 y Fk(1)p 639 114 16 2 v 639 134 a(2)634 159 y Fo(p)662 150 y Fy(\()p Fw(P)6 b(q)r Fy(\))j(+)i(\000)d(\001)884 121 y Fu(\000)916 108 y Fk(1)p 916 114 V 916 134 a(2)884 159 y Fo(p)938 150 y Fy(\()p Fw(P)e Fy(\000)p Fw(q)r Fy(\))i Fw(;)54 b(q)14 b Fv(2)f Fy(dom)8 b Fw(')1335 131 y Fu(\003)1354 150 y Fw(')13 b Fy(=)g Fw(QP)6 b Fx(q)i Fw(;)256 b Fy(\(20\))318 263 y(sgn)7 b Fw(')p Fy(\()p Fw(q)r Fy(\))41 b(=)g(\001)639 221 y Fk(1)p 639 227 V 639 248 a(2)634 272 y Fo(p)662 263 y Fy(\()p Fw(P)6 b(q)r Fy(\))j Fv(\000)i Fy(\000)d(\001)889 221 y Fk(1)p 889 227 V 889 248 a(2)884 272 y Fo(p)911 263 y Fy(\()p Fw(P)e Fy(\000)p Fw(q)r Fy(\))i Fw(;)53 b(q)15 b Fv(2)e Fx(q)8 b Fw(:)588 b Fy(\(21\))-12 357 y Fs(Mor)n(e)n(over)16 b Fy(sgn)7 b Fw(')16 b Fs(is)g(an)g(isometry)h(of)f Fx(q)h Fs(onto)f Fx(q)825 341 y Fu(?)855 357 y Fs(,)g(i.e.)g Fy(\(sgn)7 b Fw(')p Fy(\))1103 341 y Fu(\003)1122 357 y Fy(sgn)h Fw(')k Fy(=)h Fw(Q)j Fs(and)g Fy(sgn)8 b Fw(')p Fy(\(sgn)f Fw(')p Fy(\))1669 341 y Fu(\003)1701 357 y Fy(=)13 b Fw(Q)1785 341 y Fu(?)1814 357 y Fs(.)-12 455 y(Pr)n(o)n(of:)25 b Fy(F)l(rom)17 b(the)g(explicit)j(kno)o(wledge)e(of) f(all)i(the)e(domains)h(of)g(the)f(mappings)i(used)f(b)q(efore)g(it)g (is)g(easily)-12 512 y(seen)e(that)e(the)h(form)o(ulas)g(are)g(w)o (ell{de\014ned.)22 b(F)l(urther,)14 b(recall)j(Prop)q(osition)e(4.13)f (and)i(Corollary)f(4.14)e(and)-12 568 y(the)j(fact)g(that)f(the)h(t)o (w)o(o)f(terms)g(of)h(the)g(r.h.s.)f(of)g(the)h(ab)q(o)o(v)o(e)g(form)o (ulas)f(corresp)q(ond)i(to)e(the)h(decomp)q(osition)-12 625 y(of)f Fx(h)g Fy(in)h(terms)e(of)h Fw(P)6 b Fx(h)15 b Fy(and)h Fw(P)508 608 y Fu(?)538 625 y Fx(h)p Fy(.)j(Appling)e(no)o (w)e(for)f Fw(q)h Fv(2)e Fy(dom)7 b Fw(')1132 608 y Fu(\003)1152 625 y Fw(')15 b Fy(t)o(wice)g(the)g(r.h.s.)f(of)h(\(20\))f(w)o(e)h(get) 92 736 y(\001)135 694 y Fk(1)p 135 700 V 135 720 a(2)130 745 y Fo(p)157 736 y Fw(P)192 689 y FB(\020)218 736 y Fy(\001)261 694 y Fk(1)p 260 700 V 260 720 a(2)256 745 y Fo(p)283 736 y Fy(\()p Fw(P)6 b(q)r Fy(\))k(+)g(\000)e(\001)505 707 y Fu(\000)537 694 y Fk(1)p 537 700 V 537 720 a(2)505 745 y Fo(p)560 736 y Fy(\()p Fw(P)e Fy(\000)p Fw(q)r Fy(\))681 689 y FB(\021)716 736 y Fy(+)k(\000)e(\001)835 707 y Fu(\000)867 694 y Fk(1)p 867 700 V 867 720 a(2)835 745 y Fo(p)890 736 y Fw(P)e Fy(\000)953 689 y FB(\020)978 736 y Fy(\001)1021 694 y Fk(1)p 1021 700 V 1021 720 a(2)1016 745 y Fo(p)1044 736 y Fy(\()p Fw(P)g(q)r Fy(\))j(+)i(\000)d(\001)1266 707 y Fu(\000)1298 694 y Fk(1)p 1298 700 V 1298 720 a(2)1266 745 y Fo(p)1320 736 y Fy(\()p Fw(P)e Fy(\000)p Fw(q)r Fy(\))1441 689 y FB(\021)547 827 y Fy(=)42 b(\001)662 834 y Fo(p)681 827 y Fy(\()p Fw(P)6 b(q)r Fy(\))k(+)g(\000\001)895 809 y Fu(\000)p Ft(1)895 839 y Fo(p)943 827 y Fy(\()p Fw(P)c Fy(\000)p Fw(q)r Fy(\))36 b(=)1172 780 y FB(\020)1197 827 y Fw(')1227 809 y Fu(\003)1246 827 y Fw(')1276 780 y FB(\021)1301 827 y Fy(\()p Fw(q)r Fy(\))8 b Fw(;)-12 930 y Fy(whic)o(h)j(sho)o(ws)f(the)h(\014rst)f(form)o(ula.)18 b(T)l(o)10 b(pro)o(v)o(e)g(the)g(second)h(one)g(note)f(\014rst)g(that)g (for)g Fw(q)1425 914 y Fu(0)1449 930 y Fy(=)j Fv(j)p Fw(')p Fv(j)p Fy(\()p Fw(q)r Fy(\),)c Fw(q)15 b Fv(2)d Fy(dom)c Fw(')1834 914 y Fu(\003)1853 930 y Fw(')p Fy(,)-12 986 y(and)15 b(using)h(Eq.)f(\(20\))f(as)h(w)o(ell)h(as)f(Corollary)g (4.14)f(w)o(e)h(obtain)g(from)f(a)h(similar)i(calculation)g(as)d(b)q (efore)i(that)409 1051 y FB(\020)433 1098 y Fy(\001)476 1056 y Fk(1)p 476 1062 V 476 1082 a(2)471 1106 y Fo(p)498 1098 y Fw(P)h Fv(\000)11 b Fy(\000)d(\001)669 1056 y Fk(1)p 668 1062 V 668 1082 a(2)664 1106 y Fo(p)691 1098 y Fw(P)e Fy(\000)754 1051 y FB(\021)779 1098 y Fv(j)p Fw(')p Fv(j)p Fy(\()p Fw(q)875 1079 y Fu(0)886 1098 y Fy(\))12 b(=)964 1051 y FB(\020)989 1098 y Fy(\001)1027 1105 y Fo(p)1046 1098 y Fw(P)k Fv(\000)11 b Fw(P)1172 1079 y Fu(?)1202 1051 y FB(\021)1227 1098 y Fy(\()p Fw(q)1267 1079 y Fu(0)1278 1098 y Fy(\))h(=)h Fw(')p Fy(\()p Fw(q)1426 1079 y Fu(0)1437 1098 y Fy(\))8 b Fw(:)-12 1200 y Fy(Th)o(us)17 b(the)h(r.h.s.)e(and)h(the)h(l.h.s.)f(of)f(\(21\))g(coincide)k(on)d (the)g(dense)i(subspace)f(ima)7 b Fw(')1486 1183 y Fu(\003)1506 1200 y Fw(')p Fy(.)26 b(Finally)l(,)19 b(the)e(fact)-12 1273 y(that)12 b(the)h(r.h.s.)f(is)h(also)g(w)o(ell)h(de\014ned)g(for)e (all)i Fw(q)g Fv(2)f Fx(q)g Fy(\(recall)h(that)e(dom)7 b(\001)1257 1231 y Fk(1)p 1257 1237 V 1257 1258 a(2)1252 1282 y Fo(p)1292 1273 y Fy(=)13 b(dom)7 b Fw(\014)15 b Fy(=)e Fw(P)6 b Fx(q)p Fy(\))13 b(and)g(that)g(sgn)7 b Fw(')-12 1330 y Fy(maps)16 b(isometrically)j(the)d(dense)i(subspace)f (ima)8 b Fv(j)p Fw(')p Fv(j)14 b(\022)i Fx(q)h Fy(on)o(to)e(the)i (dense)h(subspace)f(ima)8 b Fw(')15 b Fy(=)g Fw(Q)1726 1313 y Fu(?)1756 1330 y Fx(p)f Fv(\022)i Fx(q)1866 1313 y Fu(?)-12 1386 y Fy(pro)o(v)o(es)g(form)o(ula)f(\(21\).)22 b(Therefore)16 b Fw(Q)h Fy(is)f(the)h(initial)h(pro)s(jection)e(of)g (sgn)7 b Fw(')16 b Fy(and)h Fw(Q)1447 1370 y Fu(?)1493 1386 y Fy(is)f(the)h(corresp)q(onding)-12 1443 y(\014nal)f(pro)s (jection.)p 1857 1443 24 24 v -12 1576 a Fq(4.16)g(Rema)o(rk)23 b Fy(Recall)18 b(that)e(dom)7 b Fw(')635 1560 y Fu(\003)655 1576 y Fw(')16 b Fy(is)g(a)g(core)g(for)g Fv(j)p Fw(')p Fv(j)f Fy(and)i(note)f(that)f(the)i(r.h.s.)e(of)h(form)o(ula)f(\(20\))g Fs(c)n(an)-12 1633 y(not)g Fy(b)q(e)h(extended)g(to)f(the)g(whole)h (dom)7 b Fv(j)p Fw(')p Fv(j)p Fy(.)59 1731 y(Finally)l(,)16 b(w)o(e)f(consider)h(the)f(mapping)600 1830 y Fx(q)e Fv(3)g Fw(q)i Fv(7!)e Fw(W)6 b(q)14 b Fy(:=)f(\(1)-11 b FF(l)15 b Fy(+)c(\001)1056 1837 y Fo(p)1075 1830 y Fy(\))1098 1797 y Fk(1)p 1098 1803 16 2 v 1098 1824 a(2)1128 1830 y Fw(P)6 b(q)15 b Fv(2)d Fx(p)c Fw(:)531 b Fy(\(22\))-12 1928 y Fq(4.17)16 b(Lemma)24 b Fw(W)f Fs(is)16 b(an)g(isometry)g(fr)n (om)h Fx(q)f Fs(onto)h Fx(p)p Fs(.)-12 2026 y(Pr)n(o)n(of:)j Fy(First)15 b(c)o(ho)q(ose)g Fw(q)403 2033 y Ft(1)423 2026 y Fw(;)8 b(q)464 2033 y Ft(2)496 2026 y Fv(2)13 b Fw(Q)p Fx(p)h Fy(whic)o(h)j(is)e(dense)h(in)g Fx(q)p Fy(.)k(No)o(w)15 b(using)247 2120 y Fv(h)p Fw(P)6 b(q)320 2127 y Ft(1)340 2120 y Fw(;)i Fy(\001)399 2127 y Fo(p)417 2120 y Fy(\()p Fw(P)e(q)490 2127 y Ft(2)510 2120 y Fy(\))p Fv(i)12 b Fy(=)h Fv(h)p Fw(P)6 b(q)679 2127 y Ft(1)700 2120 y Fw(;)i(\014)749 2101 y Fu(\003)760 2120 y Fw(\014)r Fy(\()p Fw(P)e(q)861 2127 y Ft(2)881 2120 y Fy(\))p Fv(i)12 b Fy(=)h Fv(h)p Fw(\014)r Fy(\()p Fw(P)6 b(q)1096 2127 y Ft(2)1116 2120 y Fy(\))p Fw(;)i(\014)r Fy(\()p Fw(P)e(q)1256 2127 y Ft(1)1275 2120 y Fy(\))p Fv(i)12 b Fy(=)h Fv(h)p Fw(P)1424 2101 y Fu(?)1454 2120 y Fw(q)1474 2127 y Ft(1)1494 2120 y Fw(;)8 b(P)1550 2101 y Fu(?)1579 2120 y Fw(q)1599 2127 y Ft(2)1619 2120 y Fv(i)-12 2214 y Fy(w)o(e)15 b(obtain)174 2261 y FB(D)200 2308 y Fy(\(1)-11 b FF(l)15 b Fy(+)10 b(\001)340 2315 y Fo(p)360 2308 y Fy(\))383 2275 y Fk(1)p 382 2281 V 382 2302 a(2)405 2308 y Fw(P)c(q)460 2315 y Ft(1)480 2308 y Fw(;)i Fy(\(1)-12 b FF(l)16 b Fy(+)10 b(\001)641 2315 y Fo(p)660 2308 y Fy(\))683 2275 y Fk(1)p 683 2281 V 683 2302 a(2)705 2308 y Fw(P)c(q)760 2315 y Ft(2)781 2261 y FB(E)819 2308 y Fy(=)13 b Fv(h)p Fw(P)6 b(q)940 2315 y Ft(1)960 2308 y Fw(;)i Fy(\(1)-12 b FF(l)15 b Fy(+)c(\001)1121 2315 y Fo(p)1140 2308 y Fy(\))p Fw(P)6 b(q)1213 2315 y Ft(2)1233 2308 y Fv(i)12 b Fy(=)h Fv(h)p Fw(q)1349 2315 y Ft(1)1369 2308 y Fw(;)8 b(q)1410 2315 y Ft(2)1429 2308 y Fv(i)g Fw(;)g(q)1496 2315 y Ft(1)1515 2308 y Fw(;)g(q)1556 2315 y Ft(2)1588 2308 y Fv(2)13 b Fw(Q)p Fx(p)8 b Fw(:)-12 2415 y Fy(F)l(urther,)15 b(for)f(an)o(y)h Fw(p)e Fv(2)g Fx(p)i Fy(w)o(e)g(ha)o(v)o(e)g(from)g(Prop)q(osition)g (4.12)f(that)h(\(1)-11 b FF(l)15 b Fy(+)c(\001)1298 2422 y Fo(p)1317 2415 y Fy(\)\()p Fw(P)6 b(Qp)p Fy(\))12 b(=)h Fw(P)6 b(Qp)11 b Fy(+)f Fw(P)c(Q)1746 2398 y Fu(?)1776 2415 y Fw(p)13 b Fy(=)g Fw(p)p Fy(.)-12 2480 y(This)j(implies)h(that)e (ima)7 b(\(1)-11 b FF(l)16 b Fy(+)10 b(\001)566 2487 y Fo(p)585 2480 y Fy(\))608 2448 y Fk(1)p 608 2454 V 608 2475 a(2)643 2480 y Fy(=)j Fx(p)p Fy(,)h(since)546 2583 y Fx(p)e Fy(=)h(ima)8 b(\(1)-11 b FF(l)15 b Fy(+)c(\001)851 2590 y Fo(p)870 2583 y Fy(\))h Fv(\022)h Fy(ima)8 b(\(1)-11 b FF(l)16 b Fy(+)10 b(\001)1170 2590 y Fo(p)1189 2583 y Fy(\))1212 2551 y Fk(1)p 1212 2557 V 1212 2577 a(2)1247 2583 y Fv(\022)j Fx(p)8 b Fw(:)-12 2677 y Fy(Therefore)15 b(\(22\))f(is)i(the)f(isometric)h(extension)g(of)f Fw(W)6 b Fh(\026)p Fw(Q)p Fx(p)p Fy(.)p 1857 2677 24 24 v 59 2769 a(Using)18 b(no)o(w)f(the)g(isometry)h Fw(W)24 b Fy(w)o(e)17 b(conclude)i(this)f(section)g(sho)o(wing)g(the)f(unitary)h (equiv)m(alence)j(of)c Fv(j)p Fw(')p Fv(j)-12 2843 y Fy(and)e(\001)119 2800 y Fk(1)p 119 2806 16 2 v 119 2827 a(2)114 2851 y Fo(p)141 2843 y Fy(.)919 2967 y(14)p eop %%Page: 15 15 15 14 bop -12 50 a Fq(4.18)16 b(Theo)o(rem)23 b Fs(With)17 b(the)g(pr)n(e)n(c)n(e)n(ding)e(notation)h(we)g(have)577 167 y Fw(W)e Fv(j)p Fw(')p Fv(j)p Fy(\()p Fw(q)r Fy(\))d(=)i(\001)850 125 y Fk(1)p 850 131 16 2 v 850 152 a(2)845 176 y Fo(p)880 167 y Fw(W)6 b Fy(\()p Fw(q)r Fy(\))i Fw(;)52 b(q)15 b Fv(2)e Fy(dom)7 b Fv(j)p Fw(')p Fv(j)h Fw(:)-12 273 y Fs(Pr)n(o)n(of:)20 b Fy(F)l(or)14 b(an)o(y)h Fw(q)g Fv(2)e Fy(dom)7 b Fw(')496 257 y Fu(\003)516 273 y Fw(')p Fy(,)14 b(whic)o(h)i(is)g(a)f(core)g(of)g Fv(j)p Fw(')p Fv(j)p Fy(,)e(w)o(e)i(ma)o(y)g(use)g(\(20\))f(and)i(in)g(this)f(case) 109 393 y Fw(W)e Fv(j)p Fw(')p Fv(j)p Fy(\()p Fw(q)r Fy(\))e(=)i(\(1)-11 b FF(l)16 b Fy(+)10 b(\001)479 400 y Fo(p)498 393 y Fy(\))521 361 y Fk(1)p 521 367 V 521 387 a(2)543 393 y Fw(P)578 346 y FB(\020)604 393 y Fy(\001)647 351 y Fk(1)p 646 357 V 646 377 a(2)642 401 y Fo(p)669 393 y Fy(\()p Fw(P)c(q)r Fy(\))k(+)g(\000)e(\001)891 364 y Fu(\000)923 351 y Fk(1)p 923 357 V 923 377 a(2)891 401 y Fo(p)946 393 y Fy(\()p Fw(P)e Fy(\000)p Fw(q)r Fy(\))1067 346 y FB(\021)1104 393 y Fy(=)13 b(\(1)-11 b FF(l)15 b Fy(+)c(\001)1293 400 y Fo(p)1312 393 y Fy(\))1335 361 y Fk(1)p 1335 367 V 1335 387 a(2)1365 393 y Fy(\001)1408 351 y Fk(1)p 1407 357 V 1407 377 a(2)1403 401 y Fo(p)1430 393 y Fy(\()p Fw(P)6 b(q)r Fy(\))12 b(=)h(\001)1626 351 y Fk(1)p 1626 357 V 1626 377 a(2)1621 401 y Fo(p)1648 393 y Fw(W)6 b Fy(\()p Fw(q)r Fy(\))i Fw(:)-12 527 y Fy(But)18 b(this)h(implies)h(that)e Fw(W)6 b Fv(j)p Fw(')p Fv(j)p Fw(W)586 511 y Fu(\003)623 527 y Fv(\022)18 b Fy(\001)719 485 y Fk(1)p 719 491 V 719 511 a(2)714 535 y Fo(p)759 527 y Fy(and)h(since)g(the)g(l.h.s.)f(as)f(w)o(ell)j(as)e (the)g(r.h.s.)f(of)h(the)g(preceding)-12 604 y(inclusion)g(are)c (self{adjoin)o(t)i(op)q(erators,)e(w)o(e)h(m)o(ust)f(actually)i(ha)o(v) o(e)f(the)g(equalit)o(y)h Fw(W)6 b Fv(j)p Fw(')p Fv(j)p Fw(W)1571 588 y Fu(\003)1603 604 y Fy(=)13 b(\001)1694 562 y Fk(1)p 1694 568 V 1694 589 a(2)1689 613 y Fo(p)1716 604 y Fy(.)p 1857 604 24 24 v -12 783 a Fz(5)67 b(Twisted)22 b(dualit)n(y)-6 b(.)32 b(The)23 b(generic)g(p)r(osition)g(case)-12 884 y Fy(W)l(e)13 b(b)q(egin)i(the)e(pro)q(of)f(of)h(t)o(wisted)g (dualit)o(y)h(considering)h(\014rst)d(one)i(of)e(the)i(extremal)f (cases)g(that)f(ma)o(y)g(app)q(ear)-12 941 y(in)21 b(the)g(Halmos)g (decomp)q(osition)h(\(9\).)35 b(F)l(or)20 b(this)h(assume)g(that)f(\()p Fx(h)p Fw(;)8 b Fy(\000\),)20 b Fw(P)27 b Fy(and)21 b Fw(Q)f Fy(are)h(giv)o(en)g(as)f(in)i(the)-12 997 y(preceding)d (section.)26 b(In)18 b(particular)g Fx(p)e Fy(and)i Fx(q)f Fy(are)g(in)h(generic)g(p)q(osition,)g(so)f(that)g(b)o(y)g(Prop)q (osition)g(3.4)g(the)-12 1054 y(Mo)q(dular)f(Theory)g(is)h(w)o(ell)g (de\014ned)h(for)d(the)i(pair)f(\()p Fv(M)p Fy(\()p Fx(q)p Fy(\))p Fw(;)8 b Fy(\012\).)21 b(Denote)16 b(as)g(usual)h(b)o(y)f Fw(S)g Fy(=)f Fw(J)t Fy(\001)1675 1022 y Fk(1)p 1676 1028 16 2 v 1676 1049 a(2)1714 1054 y Fy(the)h(p)q(olar)-12 1110 y(decomp)q(osition)g(of)f(the)g(T)l(omita)g(op)q(erator.)59 1175 y(W)l(e)k(pro)o(v)o(e)f(\014rst)h(that)f(the)i(di\013eren)o(t)f (mo)q(dular)h(ob)s(jects)e(lea)o(v)o(e)h(the)h Fw(n)p Fy({particle)g(submanifolds)1777 1142 y Fp(n)1772 1175 y Fv(^)q Fy(\()p Fw(P)6 b Fx(q)p Fy(\))-12 1232 y(in)o(v)m(arian)o(t.) 28 b(This)19 b(fact)e(is)h(w)o(ell)h(kno)o(wn)e(in)i(the)f(con)o(text)f 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b(that)f(the)g(\014rst)g(expression)i (holds)f(for)e(a)h(n)o(um)o(b)q(er)-12 1841 y(of)j(v)o(ectors)f Fv(\024)f Fw(n)d Fv(\000)h Fy(1.)19 b(Then)d(applying)h(this)e (induction)i(h)o(yp)q(othesis)f(as)f(w)o(ell)h(as)f(Prop)q(osition)h (2.2)e(w)o(e)h(get)97 1943 y Fw(S)s Fy(\()p Fw(P)6 b Fy(\000)p Fw(q)229 1950 y Fp(n)263 1943 y Fv(^)k Fw(:)e(:)g(:)g Fv(^)j Fw(P)6 b Fy(\000)p Fw(q)490 1950 y Ft(1)511 1943 y Fy(\))138 2081 y(=)42 b Fw(S)246 2009 y FB( )279 2081 y Fw(a)p Fy(\()p Fw(q)341 2088 y Fp(n)364 2081 y Fy(\))10 b Fv(\001)g Fw(:)e(:)g(:)g Fv(\001)h Fw(a)p Fy(\()p Fw(q)562 2088 y Ft(1)582 2081 y Fy(\))f(\012)22 b Fv(\000)763 2040 y FB(X)729 2130 y Fr(\031)10 b Fn(2)f Fm(S)813 2134 y Fl(n;p)757 2171 y Fr(p)g Fn(\025)g Fk(1)848 2081 y Fy(\(sgn)e Fw(\031)r Fy(\))1024 2025 y Fp(p)1006 2040 y FB(Y)1005 2132 y Fp(l)p Ft(=1)1073 2081 y Fv(h)p Fw(P)f(q)1146 2088 y Fp(\013)1169 2094 y Fr(l)1191 2081 y Fw(;)15 b(P)6 b Fy(\000)p Fw(q)1302 2088 y Fp(\014)1322 2094 y Fr(l)1337 2081 y Fv(i)i 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Fy(\000)p Fw(q)673 505 y Fp(n)707 498 y Fv(^)11 b Fw(:)d(:)g(:)g Fv(^)i Fw(P)c Fy(\000)p Fw(q)934 505 y Ft(1)955 498 y Fy(\))12 b(=)h Fw(P)6 b(q)1088 505 y Ft(1)1119 498 y Fv(^)k Fw(:)e(:)g(:)h Fv(^)h Fw(P)c(q)1318 505 y Fp(n)-12 576 y Fy(and)15 b(the)h(pro)q(of)e (is)i(concluded.)p 1857 576 24 24 v 59 668 a(Recall)k(from)d(Remark)h (4.11)f(\(i\))i Fw(S)s Fh(\026)p Fx(p)e Fv(\023)h Fw(\014)i Fy(and)f Fw(S)961 651 y Fu(\003)980 668 y Fh(\026)p Fx(p)e Fv(\023)h Fw(\013)p Fy(.)30 b(W)l(e)18 b(will)i(sho)o(w)e(next)g(that)g (actually)h(the)-12 735 y(equalit)o(y)g(holds.)29 b(Let)18 b Fw(P)422 742 y Fp(n)464 735 y Fy(denote)g(the)g(pro)s(jection)g(of)f Fx(F)h Fy(on)o(to)f(the)h Fw(n)p Fy({particle)h(subspace)1614 701 y Fp(n)1609 735 y Fv(^)8 b Fx(p)p Fy(.)28 b(Then)19 b(the)-12 791 y(family)13 b(of)f(orthopro)s(jections)g Fv(f)p Fw(P)563 798 y Fp(n)586 791 y Fv(g)609 798 y Fp(n)p Fu(2)p Fd(N)692 791 y Fy(is)h(m)o(utually)h(orthogonal)d(and)1235 759 y FB(P)1279 773 y Fu(1)1279 803 y Fp(n)p Ft(=0)1355 791 y Fw(P)1384 798 y Fp(n)1420 791 y Fy(=)i(1)-10 b FF(l)1498 798 y Fo(F)1521 791 y Fy(.)19 b(F)l(urther)12 b(w)o(e)h(de\014ne)-12 848 y(the)i(op)q(erator)g Fw(S)278 855 y Fk(\014n)329 848 y Fy(b)o(y)253 944 y(dom)8 b Fw(S)375 951 y Fk(\014n)453 944 y Fy(:=)41 b(span)641 897 y FB(n)668 944 y Fw(a)p Fy(\()p Fw(q)730 951 y Ft(1)750 944 y Fy(\))10 b Fv(\001)g Fw(:)e(:)g(:)g Fv(\001)h Fw(a)p Fy(\()p Fw(q)948 951 y Fp(n)972 944 y Fy(\)\012)j Fv(j)g Fw(q)1080 951 y Ft(1)1100 944 y Fw(;)c(:)g(:)g(:)d(;)j(q)1222 951 y Fp(n)1258 944 y Fv(2)13 b Fx(q)g Fw(;)19 b(n)13 b Fv(2)g Fg(N)c Fv([)i(f)p Fy(0)p Fv(g)1604 897 y FB(o)321 1018 y Fw(S)349 1025 y Fk(\014n)385 1018 y Fw(x)42 b Fy(:=)f Fw(S)s(x)8 b(;)52 b(x)12 b Fv(2)h Fy(dom)8 b Fw(S)875 1025 y Fk(\014n)919 1018 y Fw(:)-12 1108 y Fq(5.2)16 b(Lemma)24 b Fy(dom)8 b Fw(S)365 1115 y Fc(\014n)417 1108 y Fs(is)15 b(a)i(c)n(or)n(e)f(for)g(the)h(T)m(omita)f(op)n(er)n (ator)h Fw(S)s Fs(.)-12 1193 y(Pr)n(o)n(of:)k Fy(Put)16 b Fv(C)g Fy(:=)d(dom)8 b Fw(S)442 1200 y Fk(\014n)494 1193 y Fy(and)16 b(denote)g(b)o(y)g Fv(A)p Fy(\()p Fx(q)p Fy(\))f(the)h(*{algebra)g(generated)f(b)o(y)h Fv(f)p Fw(a)p Fy(\()p Fw(q)r Fy(\))d Fv(j)g Fw(q)j Fv(2)d Fx(q)p Fv(g)p Fy(,)j(so)f(that)-12 1249 y Fv(M)p Fy(\()p Fx(q)p Fy(\))d(=)h Fv(A)p Fy(\()p Fx(q)p Fy(\))255 1232 y Fu(00)276 1249 y Fy(.)19 b(F)l(urther)11 b(recall)j(that)d(if)h Fw(S)749 1256 y Ft(0)769 1249 y Fy(\()p Fw(M)5 b Fy(\012\))12 b(:=)g Fw(M)1008 1232 y Fu(\003)1028 1249 y Fy(\012,)g Fw(M)18 b Fv(2)13 b(M)p Fy(\()p Fx(q)p Fy(\),)e(then)i(the)f(graph)f (of)h(the)g(T)l(omita)-12 1305 y(op)q(erator)i Fw(S)h Fy(=)e(clo)c Fw(S)354 1312 y Ft(0)388 1305 y Fy(can)16 b(b)q(e)f(written)h(as)385 1401 y(gra)7 b Fw(S)15 b Fy(=)e(clo)603 1408 y Fp(w)631 1401 y Fy(\(gra)6 b Fw(S)747 1408 y Ft(0)767 1401 y Fy(\))12 b(=)h(clo)901 1408 y Fp(w)929 1354 y FB(n)957 1401 y Fy(\()p Fw(M)5 b Fy(\012)p Fw(;)j(M)1127 1382 y Fu(\003)1145 1401 y Fy(\012\))13 b Fv(j)f Fw(M)17 b Fv(2)c(M)p Fy(\()p Fx(q)p Fy(\))1451 1354 y FB(o)1486 1401 y Fw(;)-12 1497 y Fy(where)j(clo)176 1504 y Fp(w)220 1497 y Fy(denotes)g(the)h(closure)f(in)h(the)g(w)o(eak)e(op)q(erator)g (top)q(ology)l(.)23 b(Th)o(us)16 b(to)f(pro)o(v)o(e)h(the)g(core)g (prop)q(ert)o(y)-12 1553 y(of)f Fv(C)s Fy(,)f(i.e.)h(clo)8 b(\()p Fw(S)s Fh(\026)p Fv(C)s Fy(\))j(=)i Fw(S)s Fy(,)i(w)o(e)g(need)h (to)e(sho)o(w)h(that)300 1652 y(clo)356 1659 y Fp(w)384 1605 y FB(n)412 1652 y Fy(\()p Fw(M)5 b Fy(\012)p Fw(;)j(M)582 1633 y Fu(\003)600 1652 y Fy(\012\))k Fv(j)h Fw(M)k Fv(2)c(M)p Fy(\()p Fx(q)p Fy(\))906 1605 y FB(o)946 1652 y Fy(=)g(clo)1049 1659 y Fp(w)1077 1605 y FB(n)1105 1652 y Fy(\()p Fw(A)p Fy(\012)p Fw(;)8 b(A)1245 1633 y Fu(\003)1264 1652 y Fy(\012\))k Fv(j)g Fw(A)h Fv(2)g(A)p Fy(\()p Fx(q)p Fy(\))1536 1605 y FB(o)1571 1652 y Fw(:)59 1750 y Fy(No)o(w)h(for)h(eac)o(h)h Fw(M)i Fv(2)c(M)p Fy(\()p Fx(q)p Fy(\))g(there)i(exists)g(a)f(sequence) i Fv(f)p Fw(A)1095 1757 y Fp(n)1118 1750 y Fv(g)1141 1757 y Fp(n)p Fu(2)p Fd(N)1225 1750 y Fv(\032)d(A)p Fy(\()p Fx(q)p Fy(\))h(with)h Fv(jh)p Fw(x;)8 b Fy(\()p Fw(M)13 b Fv(\000)e Fw(A)1720 1757 y Fp(n)1744 1750 y Fy(\))p Fw(y)r Fv(ij)h Fw(<)h(\017)-12 1806 y Fy(for)i(all)h Fw(x;)8 b(y)13 b Fv(2)g Fx(F)i Fy(if)h Fw(n)f Fy(is)h(su\016cien)o(tly) h(large.)i(This)d(implies)i(that)c(for)h(all)h(\()p Fw(x;)8 b(y)r Fy(\))j Fv(2)i Fx(F)d Fv(\002)g Fx(F)p Fy(,)15 b(w)o(e)g(ha)o(v)o(e)17 1856 y FB(\014)17 1881 y(\014)17 1906 y(\014)31 1858 y(D)56 1905 y Fy(\()p Fw(M)5 b Fy(\012)p Fw(;)j(M)226 1886 y Fu(\003)245 1905 y Fy(\012\))p Fw(;)g Fy(\()p Fw(x;)g(y)r Fy(\))424 1858 y FB(E)456 1905 y Fv(\000)502 1858 y FB(D)527 1905 y Fy(\()p Fw(A)579 1912 y Fp(n)603 1905 y Fy(\012)p Fw(;)g(A)691 1886 y Fu(\003)691 1916 y Fp(n)713 1905 y Fy(\012\))p Fw(;)g Fy(\()p Fw(x;)g(y)r Fy(\))892 1858 y FB(E)915 1856 y(\014)915 1881 y(\014)915 1906 y(\014)941 1905 y Fy(=)989 1856 y FB(\014)989 1881 y(\014)989 1906 y(\014)1003 1858 y(D)1028 1905 y Fy(\()p Fw(M)15 b Fv(\000)c Fw(A)1185 1912 y Fp(n)1208 1905 y Fy(\)\012)p Fw(;)d(x)1306 1858 y FB(E)1340 1905 y Fv(\000)1386 1858 y FB(D)1411 1905 y Fy(\()p Fw(M)1478 1886 y Fu(\003)1508 1905 y Fv(\000)i Fw(A)1587 1886 y Fu(\003)1587 1916 y Fp(n)1611 1905 y Fy(\)\012)p Fw(;)e(y)1707 1858 y FB(E)1731 1856 y(\014)1731 1881 y(\014)1731 1906 y(\014)1757 1905 y Fv(\024)13 b Fy(2)p Fw(\017)8 b(;)-12 2001 y Fy(where)14 b(w)o(e)f(ha)o(v)o(e)g(used)h(that)e(in)j(the)e(w)o(eak)g(op)q(erator)f (top)q(ology)h(if)h Fw(A)1164 2008 y Fp(n)1200 2001 y Fv(!)g Fw(M)5 b Fy(,)13 b(then)h Fw(A)1470 1984 y Fu(\003)1470 2012 y Fp(n)1506 2001 y Fv(!)f Fw(M)1613 1984 y Fu(\003)1633 2001 y Fy(.)19 b(Therefore)13 b(if)-12 2068 y(\()p Fw(M)50 2075 y Fp(l)63 2068 y Fy(\012)p Fw(;)8 b(M)166 2052 y Fu(\003)161 2082 y Fp(l)185 2068 y Fy(\012\))279 2043 y Fp(l)248 2068 y Fv(\000)-8 b(!)14 b Fy(\()p Fw(x)378 2075 y Ft(0)397 2068 y Fw(;)8 b(y)440 2075 y Ft(0)459 2068 y Fy(\))14 b(w)o(eakly)l(,)h(then)g(for)e(eac)o(h)i Fw(l)g Fy(there)g(exists)f(a)h(sequence)g Fv(f)p Fw(A)1474 2078 y Ft(\()p Fp(n;l)p Ft(\))1546 2068 y Fv(g)1569 2075 y Fp(n)p Fu(2)p Fd(N)1652 2068 y Fv(\032)e(A)p Fy(\()p Fx(q)p Fy(\))h(suc)o(h)-12 2134 y(that)g(\()p Fw(A)138 2143 y Ft(\()p Fp(n;l)p Ft(\))210 2134 y Fy(\012)p Fw(;)8 b(A)298 2117 y Fu(\003)298 2149 y Ft(\()p Fp(n;l)p Ft(\))369 2134 y Fy(\012\))458 2109 y Fp(n)432 2134 y Fv(\000)-7 b(!)13 b Fy(\()p Fw(M)580 2141 y Fp(l)592 2134 y Fy(\012)p Fw(;)8 b(M)695 2117 y Fu(\003)690 2147 y Fp(l)714 2134 y Fy(\012\))15 b(w)o(eakly)l(.)20 b(Finally)l(,)c(the)g(estimate)155 2195 y FB(\014)155 2220 y(\014)155 2245 y(\014)169 2197 y(D)194 2244 y Fy(\()p Fw(A)246 2253 y Ft(\()p Fp(n;l)p Ft(\))318 2244 y Fy(\012)p Fw(;)8 b(A)406 2225 y Fu(\003)406 2257 y Ft(\()p Fp(n;l)p Ft(\))476 2244 y Fy(\012\))p Fw(;)g Fy(\()p Fw(x;)g(y)r Fy(\))655 2197 y FB(E)688 2244 y Fv(\000)734 2197 y FB(D)759 2244 y Fy(\()p Fw(x)803 2251 y Ft(0)822 2244 y Fw(;)g(y)865 2251 y Ft(0)884 2244 y Fy(\))p Fw(;)g Fy(\()p Fw(x;)g(y)r Fy(\))1030 2197 y FB(E)1053 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Fy(\))p Fw(;)g Fy(\()p Fw(x;)g(y)r Fy(\))1280 2379 y FB(E)1303 2378 y(\014)1303 2403 y(\014)1303 2428 y(\014)-12 2522 y Fy(completes)16 b(the)f(pro)q(of.)p 1857 2522 V 59 2614 a(F)l(rom)f(Prop)q(ositions)i(5.1)e(and)h(2.2)f(w)o (e)h(obtain)h(further)472 2705 y Fw(P)501 2712 y Fp(n)532 2705 y Fy(dom)8 b Fw(S)654 2712 y Fk(\014n)732 2705 y Fv(\022)41 b Fy(dom)8 b Fw(S)930 2712 y Fk(\014n)974 2705 y Fw(;)52 b(n)13 b Fy(=)g(0)p Fw(;)8 b Fy(1)p Fw(;)g(:)g(:)g(:)j (;)472 2774 y(P)501 2781 y Fp(n)532 2774 y Fy(dom)d Fw(S)654 2781 y Fk(\014n)732 2774 y Fv(\032)41 b Fw(P)837 2781 y Fp(n)869 2774 y Fx(F)25 b Fy(is)15 b(dense)9 b Fw(;)52 b(n)13 b Fy(=)g(0)p Fw(;)8 b Fy(1)p Fw(;)g(:)g(:)g(:)j(;)505 2843 y(P)534 2850 y Fp(n)565 2843 y Fw(S)593 2850 y Fk(\014n)637 2843 y Fw(P)666 2850 y Fp(n)732 2843 y Fy(=)41 b Fw(S)836 2850 y Fk(\014n)880 2843 y Fw(P)909 2850 y Fp(n)941 2843 y Fw(;)52 b(n)13 b Fy(=)g(0)p Fw(;)8 b Fy(1)p Fw(;)g(:)g(:)f(:)12 b(:)919 2967 y Fy(16)p eop %%Page: 17 17 17 16 bop -12 50 a Fy(Note)15 b(that)g(for)g Fw(x)f Fv(2)f Fy(dom)8 b Fw(S)472 57 y Fk(\014n)524 50 y Fy(the)15 b(series)i Fw(x)c Fy(=)813 18 y FB(P)857 31 y Fu(1)857 62 y Fp(n)p Ft(=0)933 50 y Fw(P)962 57 y Fp(n)985 50 y Fw(x)j Fy(is)g(a)g(\014nite)g(sum.)21 b(F)l(urther)16 b(w)o(e)f(consider)i(the)f(op-)-12 107 y(erators)e Fw(S)168 114 y Fk(\014n)205 107 y Fy(\()p Fw(n)p Fy(\))e(:)h Fw(P)335 114 y Fp(n)359 107 y Fx(F)g Fv(!)g Fw(P)487 114 y Fp(n)511 107 y Fx(F)i Fy(b)o(y)h(dom)7 b Fw(S)739 114 y Fk(\014n)776 107 y Fy(\()p Fw(n)p Fy(\))12 b(:=)i Fw(P)942 114 y Fp(n)973 107 y Fy(dom)7 b Fw(S)1094 114 y Fk(\014n)1146 107 y Fy(and)16 b Fw(S)1263 114 y Fk(\014n)1299 107 y Fy(\()p Fw(n)p Fy(\))p Fw(x)d Fy(:=)g Fw(S)1490 114 y Fk(\014n)1526 107 y Fw(x)8 b(;)g(x)k Fv(2)i Fy(dom)7 b Fw(S)1784 114 y Fk(\014n)1820 107 y Fy(\()p Fw(n)p Fy(\).)-12 163 y(Recall)17 b(from)e(Lemma)g(4.10)f(that)h Fw(S)618 170 y Fk(\014n)654 163 y Fy(\(1\))d(=)h Fw(\014)k Fy(is)f(a)f(closed)i(densely)f (de\014ned)h(op)q(erator)d(from)h Fx(p)e Fy(=)g Fw(P)1756 170 y Ft(1)1776 163 y Fx(F)i Fy(in)o(to)-12 220 y Fx(p)g Fy(and)g(b)o(y)g(the)h(preceding)g(argumen)o(ts)f(w)o(e)g(ha)o(v)o(e) 505 331 y Fw(S)533 338 y Fk(\014n)582 331 y Fy(=)645 297 y Fu(1)645 331 y Fv(\010)630 364 y Fp(n)p Ft(=0)704 331 y Fw(S)732 338 y Fk(\014n)768 331 y Fy(\()p Fw(n)p Fy(\))90 b(\(algebraic)16 b(direct)g(sum\))7 b Fw(:)436 b Fy(\(25\))-12 449 y(W)l(e)15 b(can)h(no)o(w)e(state:)-12 542 y Fq(5.3)i(Prop)q(osition)24 b Fs(The)19 b(T)m(omita)g(op)n(er)n (ator)g Fw(S)j Fs(and)c(its)h(adjoint)g Fw(S)1161 526 y Fu(\003)1199 542 y Fs(c)n(an)g(b)n(e)f(r)n(estricte)n(d)g(to)i(the)f Fw(n)p Fs({p)n(articles)-12 599 y(subsp)n(ac)n(es)c Fw(P)218 606 y Fp(n)242 599 y Fx(F)p Fs(.)20 b(We)d(have)305 701 y Fw(S)s Fh(\026)p Fw(P)384 708 y Fp(n)407 701 y Fx(F)12 b Fy(=)h(clo)c Fw(S)587 708 y Fc(\014n)622 701 y Fy(\()p Fw(n)p Fy(\))46 b(and)h Fw(S)882 682 y Fu(\003)901 701 y Fh(\026)p Fw(P)949 708 y Fp(n)972 701 y Fx(F)13 b 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b(=)h(gra)7 b(clo)h Fw(S)680 1175 y Fk(\014n)716 1168 y Fy(\()p Fw(n)p Fy(\),)15 b Fw(n)e Fv(\025)g Fy(2)i(and)g(that)f (no)o(w)h(the)h(direct)f(sum)h(\(26\),)d(in)j(con)o(trast)e(to)-12 1224 y(\(25\),)i(is)i(the)g(Hilb)q(ert)h(sum)e(of)h(these)f(subspaces.) 28 b(Eq.)17 b(\(26\))f(implies)j(immediately)h(the)d(assertion)g(that)g Fw(S)-12 1281 y Fy(can)d(b)q(e)g(restricted)f(to)g Fw(P)413 1288 y Fp(n)437 1281 y Fx(F)g Fy(and)h(that)f(the)g(restriction)h (coincides)i(with)d(clo)8 b Fw(S)1337 1288 y Fk(\014n)1374 1281 y Fy(\()p Fw(n)p Fy(\),)13 b(in)h(particular)g Fw(S)s Fh(\026)p Fx(p)e Fy(=)h Fw(\014)r Fy(.)-12 1337 y(The)i(statemen)o(ts)f (concerning)j Fw(S)569 1321 y Fu(\003)603 1337 y Fy(are)e(sho)o(wn)g (analogously)l(.)p 1857 1337 24 24 v -12 1479 a Fq(5.4)h(Co)o(rolla)o (ry)23 b Fs(L)n(et)d Fw(S)i Fy(=)e Fw(J)t Fy(\001)534 1447 y Fk(1)p 535 1453 16 2 v 535 1474 a(2)577 1479 y Fs(b)n(e)g(the)h(p)n(olar)f(de)n(c)n(omp)n(osition)g(of)g(the)h(T)m (omita)f(op)n(er)n(ator.)33 b(The)20 b(mo)n(dular)-12 1535 y(op)n(er)n(ator)f Fy(\001)d(=)h Fw(S)306 1519 y Fu(\003)325 1535 y Fw(S)k Fs(and)e(the)f(mo)n(dular)i(c)n(onjugation)e Fw(J)23 b Fs(c)n(an)17 b(b)n(e)h(r)n(estricte)n(d)g(to)h(the)f(r)n(esp) n(e)n(ctive)f Fw(n)p Fs({p)n(article)-12 1592 y(subsp)n(ac)n(es.)i(In)d (p)n(articular)h(we)f(have:)28 1685 y(\(i\))22 b(Mo)n(dular)29 b(op)n(er)n(ator:)47 b Fy(\001)p Fh(\026)p Fx(p)36 b Fy(=)h(\001)743 1692 y Fo(p)791 1685 y Fs(\(r)n(e)n(c)n(al)r(l)28 b(fr)n(om)h(the)g(pr)n(e)n(c)n(e)n(ding)f(se)n(ction)g(that)i Fy(\001)1668 1692 y Fo(p)1723 1685 y Fy(=)37 b Fw(\014)1823 1669 y Fu(\003)1835 1685 y Fw(\014)r Fs(\),)102 1755 y Fy(dom)7 b(\001)p Fh(\026)p Fw(P)281 1762 y Fp(n)304 1755 y Fx(F)13 b Fy(=)397 1721 y Fp(n)393 1755 y Fv(^)8 b Fy(dom)f(\001)562 1762 y Fo(p)598 1755 y Fs(and)266 1857 y Fy(\001\()p Fw(p)345 1864 y Ft(1)374 1857 y Fv(^)k Fw(:)d(:)g(:)g Fv(^)i Fw(p)541 1864 y Fp(n)564 1857 y Fy(\))j(=)g(\(\001)699 1864 y Fo(p)725 1857 y Fw(p)748 1864 y Ft(1)768 1857 y 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Fv(^)j Fw(P)6 b(q)472 1105 y Fp(n)506 1098 y Fv(^)11 b Fy(\001)590 1056 y Fk(1)p 589 1062 V 589 1083 a(2)585 1107 y Fo(p)612 1098 y Fw(P)6 b(q)12 b Fy(+)745 1045 y Fp(n)726 1058 y FB(X)725 1149 y Fp(r)q Ft(=1)794 1098 y Fy(\()p Fv(\000)p Fy(1\))888 1080 y Fp(n)p Fu(\000)p Fp(r)963 1098 y Fv(h)p Fy(\001)1024 1056 y Fk(1)p 1024 1062 V 1024 1083 a(2)1019 1107 y Fo(p)1046 1098 y Fw(P)6 b Fy(\000)p Fw(q)r(;)i(P)e(q)1207 1105 y Fp(r)1227 1098 y Fv(i)12 b Fw(P)6 b(q)1312 1105 y Ft(1)1343 1098 y Fv(^)k Fw(:)e(:)g(:)g Fv(^)1494 1085 y FB(d)1487 1098 y Fw(P)e(q)1542 1105 y Fp(r)1571 1098 y Fv(^)11 b Fw(:)d(:)g(:)g Fv(^)i Fw(P)c(q)1770 1105 y Fp(n)1802 1098 y Fw(;)-12 1245 y Fy(where)20 b(for)f(the)g(last)h(equation)g(w)o (e)f(ha)o(v)o(e)h(used)g(that)f Fv(\000)p Fw(i)p 1004 1206 87 2 v(\021)r Fy(\()p Fw(n)p Fy(\))o Fw(\021)r Fy(\()p Fw(n)12 b Fy(+)h(1\))20 b(=)g Fv(\000)p Fw(i)p 1388 1206 V(\021)r Fy(\()p Fw(n)p Fy(\))o Fw(\021)r Fy(\()p Fw(n)12 b Fv(\000)h Fy(1\))20 b(=)g(\()p Fv(\000)p Fy(1\))1815 1228 y Fp(n)p Ft(+1)1883 1245 y Fy(.)-12 1319 y(Finally)l(,)c(the)g (equalit)o(y)g(of)e(b)q(oth)i(sides)g(follo)o(ws)f(from)g(the)g(fact)g (that)f(\001)1237 1276 y Fk(1)p 1237 1282 16 2 v 1237 1303 a(2)1232 1327 y Fo(p)1259 1319 y Fw(P)6 b(q)15 b Fy(=)e Fw(J)t(S)s(P)6 b(q)15 b Fy(=)e Fw(J)t(P)6 b Fy(\000)p Fw(q)r Fy(,)16 b Fw(q)f Fv(2)e Fx(q)p Fy(.)p 1857 1319 24 24 v -12 1460 a Fq(5.7)j(Rema)o(rk)23 b Fy(As)17 b(exp)q(ected)g(w)o (e)f(can)h(relate)g(the)f(mapping)h Fw(V)26 b Fy(de\014ned)18 b(b)q(efore)f(with)g(the)f(mapping)h Fw(j)i Fy(\(the)-12 1517 y(mo)q(dular)f(conjugation)g(restricted)g(to)f(the)h(one{particle) h(Hilb)q(ert)g(space\))f(used)g(in)h([20)o(,)f(p.)f(738].)27 b(Indeed,)-12 1573 y(considering)17 b(the)e(pro)s(jection)g(on)o(to)f (the)i(one{particle)g(Hilb)q(ert)g(space)g(w)o(e)f(ha)o(v)o(e)g(the)g (relation)416 1690 y Fw(J)t Fy(\()p Fw(P)6 b(q)r Fy(\))13 b(=)g(\001)642 1648 y Fk(1)p 641 1654 16 2 v 641 1675 a(2)637 1699 y Fo(p)664 1690 y Fw(S)s Fy(\()p Fw(P)6 b(q)r Fy(\))12 b(=)h(\001)891 1648 y Fk(1)p 890 1654 V 890 1675 a(2)886 1699 y Fo(p)913 1690 y Fy(\()p Fw(P)6 b Fy(\000)p Fw(q)r Fy(\))13 b(=)g Fv(\000)p Fw(iP)6 b(V)k Fy(\()p Fw(q)r Fy(\))e Fw(;)51 b(q)15 b Fv(2)e Fx(q)8 b Fw(:)-12 1797 y Fq(5.8)16 b(Theo)o(rem)23 b Fs(\(Twiste)n(d)14 b(Duality\))h(L)n(et)f Fv(M)p Fy(\()p Fx(q)p Fy(\))g Fs(b)n(e)h(the)g(von)g(Neumann)f(algebr)n(a)h(given)f(at)h(the)g(b)n(e) n(ginning)e(of)-12 1853 y(this)j(se)n(ction.)k(Then)717 1909 y Fv(M)p Fy(\()p Fx(q)p Fy(\))830 1891 y Fu(0)853 1909 y Fy(=)911 1896 y FB(e)901 1909 y Fw(Z)11 b Fv(M)p Fy(\()p Fx(q)1038 1891 y Fu(?)1067 1909 y Fy(\))1102 1896 y FB(e)1093 1909 y Fw(Z)1127 1891 y Fu(\003)1154 1909 y Fw(:)-12 2016 y Fs(Pr)n(o)n(of:)20 b Fy(F)l(rom)14 b(the)i(last)f(theorem)g(and)g(using)h(standard)f(results)g(in)h(Mo)q (dular)g(Theory)f(w)o(e)g(ha)o(v)o(e)339 2118 y Fv(M)p Fy(\()p Fx(q)p Fy(\))452 2099 y Fu(0)504 2118 y Fy(=)42 b Fw(J)12 b Fv(M)p Fy(\()p Fx(q)p Fy(\))c Fw(J)40 b Fy(=)c Fw(J)12 b Fv(f)p Fw(a)p Fy(\()p Fw(q)r Fy(\))g Fv(j)g Fw(q)j Fv(2)e Fx(q)p Fv(g)1177 2099 y Fu(00)1211 2118 y Fw(J)504 2210 y Fy(=)42 b Fv(f)p Fw(J)12 b(a)p Fy(\()p Fw(q)r Fy(\))c Fw(J)16 b Fv(j)c Fw(q)j Fv(2)e Fx(q)p Fv(g)920 2192 y Fu(00)504 2303 y Fy(=)42 b Fv(f)613 2290 y FB(e)604 2303 y Fw(Z)10 b(a)p Fy(\()p Fw(V)g(q)r Fy(\))780 2290 y FB(e)771 2303 y Fw(Z)805 2284 y Fu(\003)838 2303 y Fv(j)i Fw(q)i Fv(2)f Fx(q)p Fv(g)985 2284 y Fu(00)1043 2303 y Fy(=)1124 2290 y FB(e)1114 2303 y Fw(Z)e Fv(f)p Fw(a)p Fy(\()p Fw(V)e(q)r Fy(\))j Fv(j)h Fw(q)h Fv(2)f Fx(q)p Fv(g)1457 2284 y Fu(00)1500 2290 y FB(e)1491 2303 y Fw(Z)1525 2284 y Fu(\003)504 2395 y Fy(=)590 2382 y FB(e)581 2395 y Fw(Z)e Fv(M)p Fy(\()p Fx(q)718 2377 y Fu(?)747 2395 y Fy(\))781 2382 y FB(e)772 2395 y Fw(Z)806 2377 y Fu(\003)834 2395 y Fw(;)-12 2498 y Fy(where)k(w)o(e)g(ha)o(v)o (e)g(used)h(that)e Fw(V)25 b Fy(is)16 b(an)f(an)o(ti{linear)h(isometry) f(from)g Fx(q)g Fy(on)o(to)f Fx(q)1335 2481 y Fu(?)1365 2498 y Fy(.)p 1857 2498 24 24 v -12 2639 a Fq(5.9)i(Rema)o(rk)23 b Fy(If)15 b(one)g(do)q(es)f(not)g(w)o(an)o(t)g(to)g(b)q(other)g(ab)q (out)h(domain)f(questions,)h(there)g(is)g(an)f(alternativ)o(e)h(w)o(a)o (y)-12 2696 y(to)f(sho)o(w)h(the)g(preceding)h(result.)21 b(Indeed,)16 b(one)f(can)g(\014rst)g(pro)o(v)o(e)f(the)h(statemen)o(ts) f(in)i(this)g(section)f(for)g(\014nite)-12 2752 y(dimensional)j (subspaces)f Fx(q)470 2759 y Fp(n)509 2752 y Fy(of)f Fx(q)g Fy(and)g(then)g(apply)h(the)f(AFD{prop)q(ert)o(y)f(of)g Fv(M)p Fy(\()p Fx(q)p Fy(\))g(as)h(in)h(the)f(pro)q(of)f(of)h([9)o(,) -12 2809 y(Theorem)f(15.1.3].)919 2967 y(18)p eop %%Page: 19 19 19 18 bop -12 50 a Fz(6)67 b(Twisted)22 b(dualit)n(y)-6 b(.)32 b(The)23 b(ab)r(elian)g(case)-12 152 y Fy(W)l(e)13 b(will)j(pro)o(v)o(e)c(in)j(the)e(presen)o(t)h(section)g(the)f(t)o (wisted)h(dualit)o(y)g(condition)h(in)f(the)f(other)g(extremal)h(case)f (that)-12 208 y(ma)o(y)f(app)q(ear)h(in)g(the)g(Halmos)f(decomp)q (osition)i(\(9\).)k(Let)13 b(\()p Fx(h)p Fw(;)8 b Fy(\000\))j(and)i Fw(P)q(;)8 b(Q)k Fy(with)h(\000)p Fw(P)6 b Fy(\000)14 b(=)f Fw(P)1578 192 y Fu(?)1621 208 y Fy(and)f(\000)p Fw(Q)h Fy(=)g Fw(Q)p Fy(\000)-12 265 y(b)q(e)k(giv)o(en)g(as)g(in)g (Subsection)h(2.2)e(and)g(consider)i(the)f(complemen)o(tary)f (situation)h(to)f(the)h(generic)h(p)q(osition)-12 321 y(case,)d(namely)840 377 y Fw(P)6 b(Q)13 b Fy(=)g Fw(QP)-12 461 y Fy(\(i.e.)19 b(in)h(the)f(notation)g(of)f(Section)i(3)f(w)o(e)g (ha)o(v)o(e)g(no)o(w)f(that)h Fx(h)1073 468 y Ft(1)1111 461 y Fy(=)h Fv(f)p Fy(0)p Fv(g)p Fy(\).)30 b(In)20 b(this)f(situation) h(w)o(e)f(ma)o(y)f(no)o(w)-12 517 y(consider)e(the)f(follo)o(wing)h (restrictions)g(to)e Fx(q)f Fy(=)g Fw(Q)p Fx(h)i Fy(and)g Fx(q)994 501 y Fu(?)1037 517 y Fy(=)e Fw(Q)1121 501 y Fu(?)1150 517 y Fx(h)p Fy(:)427 620 y Fw(P)456 627 y Fo(q)488 620 y Fy(:=)g Fw(P)6 b Fh(\026)p Fx(q)i Fw(;)20 b Fy(\000)694 627 y Fo(q)726 620 y Fy(:=)12 b(\000)p Fh(\026)p Fx(q)c Fw(;)20 b(P)925 631 y Fo(q)942 621 y Fn(?)983 620 y Fy(:=)13 b Fw(P)6 b Fh(\026)p Fx(q)1120 601 y Fu(?)1158 620 y Fw(;)19 b Fy(\000)1218 631 y Fo(q)1235 621 y Fn(?)1276 620 y Fy(:=)13 b(\000)p Fh(\026)p Fx(q)1406 601 y Fu(?)1444 620 y Fw(:)-12 726 y Fy(Note)g(that)h Fw(P)223 733 y Fo(q)255 726 y Fy(and)g Fw(P)371 737 y Fo(q)388 728 y Fn(?)430 726 y Fy(are)g(basis)g(pro)s(jections)g(for)f (\()p Fx(q)p Fw(;)8 b Fy(\000)1005 733 y Fo(q)1023 726 y Fy(\))14 b(and)g(\()p Fx(q)1182 710 y Fu(?)1211 726 y Fw(;)8 b Fy(\000)1260 737 y Fo(q)1277 728 y Fn(?)1305 726 y Fy(\),)13 b(resp)q(ectiv)o(ely)l(.)22 b(W)l(e)13 b(denote)i(the)-12 783 y(corresp)q(onding)h(F)l(o)q(c)o(k)f(represen)o (tations)h(b)o(y)f Fw(\031)793 790 y Fo(q)827 783 y Fy(and)h Fw(\031)942 794 y Fo(q)959 784 y Fn(?)1002 783 y Fy(whic)o(h)g(act)f (on)g(the)h(an)o(ti-symmetric)f(F)l(o)q(c)o(k)g(spaces)-12 844 y Fx(F)16 851 y Fo(q)48 844 y Fy(resp.)20 b Fx(F)190 855 y Fo(q)207 845 y Fn(?)248 844 y Fy(o)o(v)o(er)13 b(the)g(one{particle)i(spaces)e Fx(p)7 b Fv(\\)g Fx(q)14 b Fy(resp.)19 b Fx(p)7 b Fv(\\)g Fx(q)1116 827 y Fu(?)1145 844 y Fy(.)20 b(Therefore)13 b(w)o(e)g(can)h(no)o(w)f(directly)i(apply) -12 905 y(Prop)q(osition)h(2.4)e(to)g(the)i(follo)o(wing)g(decomp)q (osition)g(of)f(the)g(reference)h(space)g Fx(h)c Fy(=)h Fx(q)d Fv(\010)h Fx(q)1558 888 y Fu(?)1587 905 y Fy(.)-12 1011 y Fq(6.1)16 b(Theo)o(rem)23 b Fs(\(Twiste)n(d)17 b(Duality\))g(L)n(et)g Fy(\()p Fx(h)p Fw(;)8 b Fy(\000\))16 b Fs(and)h Fw(P)q(;)8 b(Q)17 b Fs(with)h Fw(QP)j Fy(=)14 b Fw(P)6 b(Q)18 b Fs(b)n(e)f(given)g(as)g(ab)n(ove.)24 b(F)m(urther)-12 1068 y(let)17 b Fv(M)p Fy(\()p Fx(q)p Fy(\))f Fs(and)281 1054 y FB(e)272 1068 y Fw(Z)k Fs(b)n(e)d(the)h(von)f (Neumann)g(algebr)n(a)g(and)g(the)h(twist)f(op)n(er)n(ator)h(sp)n(e)n (ci\014e)n(d)d(in)i(Subse)n(ction)f(2.2.)-12 1124 y(Then)717 1181 y Fv(M)p Fy(\()p Fx(q)p Fy(\))830 1162 y Fu(0)853 1181 y Fy(=)911 1167 y FB(e)901 1181 y Fw(Z)11 b Fv(M)p Fy(\()p Fx(q)1038 1162 y Fu(?)1067 1181 y Fy(\))1102 1167 y FB(e)1093 1181 y Fw(Z)1127 1162 y Fu(\003)1154 1181 y Fw(:)648 b Fy(\(30\))-12 1287 y Fs(Pr)n(o)n(of:)22 b Fy(Note)16 b(\014rst)g(that)f(from)h(Prop)q(osition)h(2.4)e(and)h (the)h(irreducibili)q(t)o(y)h(of)e(the)h(F)l(o)q(c)o(k)f(represen)o (tation)g(w)o(e)-12 1343 y(ha)o(v)o(e)269 1400 y Fv(M)p Fy(\()p Fx(q)p Fy(\))c(=)h Fv(f)p Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)r Fy(\)\))e Fv(j)h Fw(q)j Fv(2)d Fx(q)p Fv(g)769 1381 y Fu(00)803 1400 y Fy(=)h Fv(f)p Fw(\031)900 1407 y Fo(q)919 1400 y Fy(\()p Fw(a)p Fy(\()p Fw(q)r Fy(\)\))c Fv(\012)h Fy(1)-11 b FF(l)18 b Fv(j)12 b Fw(q)j Fv(2)e Fx(q)p Fv(g)1281 1381 y Fu(00)1315 1400 y Fy(=)g Fv(L)p Fy(\()p Fx(F)1440 1407 y Fo(q)1458 1400 y Fy(\))d Fv(\012)h Fg(C)c Fy(1)-8 b FF(l)13 b Fw(:)-12 1483 y Fy(Therefore)i(using)h(again)f(standard)g(results)g(from)g([25)o(,)g (Section)h(11.2])d(w)o(e)i(obtain)h(for)e(the)i(l.h.s.)f(of)f(\(30\)) 711 1585 y Fv(M)p Fy(\()p Fx(q)p Fy(\))824 1567 y Fu(0)847 1585 y Fy(=)f Fg(C)7 b Fy(1)-8 b FF(l)16 b Fv(\012)10 b(L)p Fy(\()p Fx(F)1090 1596 y Fo(q)1107 1587 y Fn(?)1135 1585 y Fy(\))e Fw(:)-12 1687 y Fy(W)l(e)20 b(sho)o(w)f(no)o(w)h(that)f (the)h(same)f(result)h(is)h(true)f(for)f(the)h(r.h.s.)e(of)i(\(30\).)33 b(Recall)21 b(from)e(Prop)q(osition)i(2.4)-12 1744 y(that)d(the)i (implemen)o(ter)g Fw(Z)j Fy(of)c(the)g(ev)o(en{o)q(ddness)h (automorphism)f(of)g(CAR\()p Fx(h)p Fw(;)8 b Fy(\000\))18 b(on)h Fx(F)1599 1751 y Fo(q)1631 1744 y Fv(\012)13 b Fx(F)1707 1755 y Fo(q)1724 1745 y Fn(?)1771 1744 y Fy(is)20 b(just)-12 1800 y(giv)o(en)14 b(b)o(y)f Fw(Z)j Fy(=)d Fw(Z)292 1807 y Fo(q)318 1800 y Fv(\012)7 b Fw(Z)391 1811 y Fo(q)408 1802 y Fn(?)436 1800 y Fy(,)13 b(where)h Fw(Z)623 1807 y Fo(q)655 1800 y Fy(and)g Fw(Z)773 1811 y Fo(q)790 1802 y Fn(?)831 1800 y Fy(implemen)o(t)h(this)f (automorphism)f(of)g(the)h(smaller)g(algebras.)-12 1857 y(Therefore)h(w)o(e)g(get)60 1956 y FB(e)51 1969 y Fw(Z)c Fv(M)p Fy(\()p Fx(q)188 1951 y Fu(?)217 1969 y Fy(\))251 1956 y FB(e)242 1969 y Fw(Z)276 1951 y Fu(\003)338 1969 y Fy(=)414 1922 y FB(n)442 1969 y Fw(Z)s(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)586 1951 y Fu(?)615 1969 y Fy(\)\))h Fv(j)g Fw(q)710 1951 y Fu(?)752 1969 y Fv(2)h Fx(q)817 1951 y Fu(?)847 1922 y FB(o)874 1934 y Fu(00)908 1969 y Fy(=)956 1922 y FB(n\020)1009 1969 y Fw(Z)1040 1976 y Fo(q)1069 1969 y Fv(\012)d Fw(Z)1145 1981 y Fo(q)1162 1971 y Fn(?)1190 1922 y FB(\021\020)1240 1969 y Fw(Z)1271 1976 y Fo(q)1300 1969 y Fv(\012)g Fw(\031)1371 1981 y Fo(q)1388 1971 y Fn(?)1416 1969 y Fy(\()p Fw(a)p Fy(\()p Fw(q)1498 1951 y Fu(?)1527 1969 y Fy(\)\))1563 1922 y FB(\021)1600 1969 y Fv(j)i Fw(q)1647 1951 y Fu(?)1689 1969 y Fv(2)h Fx(q)1754 1951 y Fu(?)1784 1922 y FB(o)1812 1934 y Fu(00)338 2061 y Fy(=)41 b Fg(C)7 b Fy(1)-7 b FF(l)15 b Fv(\012)532 2014 y FB(\020)565 2048 y(g)557 2061 y Fw(Z)588 2072 y Fo(q)605 2062 y Fn(?)646 2061 y Fv(f)p Fw(\031)695 2072 y Fo(q)712 2062 y Fn(?)739 2061 y Fy(\()p Fw(a)p Fy(\()p Fw(q)821 2042 y Fu(?)850 2061 y Fy(\)\))d Fv(j)g Fw(q)945 2042 y Fu(?)988 2061 y Fv(2)g Fx(q)1052 2042 y Fu(?)1082 2061 y Fv(g)1105 2042 y Fu(00)1147 2048 y FB(g)1139 2061 y Fw(Z)1170 2072 y Fo(q)1187 2062 y Fn(?)1215 2030 y Fu(\003)1235 2014 y FB(\021)1272 2061 y Fy(=)h Fg(C)7 b Fy(1)-8 b FF(l)15 b Fv(\012)c(L)p Fy(\()p Fx(F)1515 2072 y Fo(q)1532 2062 y Fn(?)1560 2061 y Fy(\))d Fw(;)-12 2181 y Fy(where)131 2168 y FB(g)123 2181 y Fw(Z)154 2192 y Fo(q)171 2183 y Fn(?)217 2181 y Fy(is)19 b(the)g(t)o(wist)f(op)q(erator)g(on)g Fx(F)745 2192 y Fo(q)762 2183 y Fn(?)809 2181 y Fy(and)g(w)o(e)h(ha)o (v)o(e)f(used)h(again)g(the)f(irreducibil)q(it)o(y)j(of)d(the)h(F)l(o)q (c)o(k)-12 2237 y(represen)o(tation.)p 1857 2237 24 24 v -12 2416 a Fz(7)67 b(Twisted)22 b(dualit)n(y)-6 b(.)32 b(The)23 b(general)f(case)-12 2517 y Fy(W)l(e)16 b(are)f(no)o(w)h(in)g (a)g(p)q(osition)g(to)g(giv)o(e)g(the)g(pro)q(of)f(of)g(t)o(wisted)h (dualit)o(y)h(in)f(the)g(most)f(general)h(situation.)22 b(Let)-12 2574 y(\()p Fx(h)p Fw(;)8 b Fy(\000\))k(b)q(e)j(a)e(Hilb)q (ert)j(space)e(with)g(an)o(ti{unitary)g(in)o(v)o(olution)h(\000,)f Fw(P)20 b Fs(any)14 b Fy(basis)g(pro)s(jection)g(and)g Fx(q)g Fs(any)g Fy(closed)-12 2630 y(\000{in)o(v)m(arian)o(t)k (subspace)f(in)h Fx(h)p Fy(,)f(to)g(whic)o(h)h(w)o(e)e(asso)q(ciate)h (the)h(orthopro)s(jection)e Fw(Q)p Fy(.)25 b(F)l(rom)16 b(Lemma)h(3.1)f(and)-12 2687 y(Remark)c(3.2)g(w)o(e)g(can)g(consider)h (the)g(follo)o(wing)g(restrictions)g(w.r.t.)d(the)i(Halmos)h(decomp)q (ostion)g Fx(h)f Fy(=)h Fx(h)1789 2694 y Ft(0)1813 2687 y Fv(\010)t Fx(h)1876 2694 y Ft(1)-12 2743 y Fy(giv)o(en)j(in)g(Eq.)e (\(9\):)614 2800 y Fx(q)636 2807 y Fp(k)670 2800 y Fy(:=)f Fw(Q)p Fh(\026)p Fx(h)810 2807 y Fp(k)876 2800 y Fy(and)46 b Fx(q)1017 2781 y Fu(?)1017 2811 y Fp(k)1059 2800 y Fy(:=)13 b Fw(Q)1156 2781 y Fu(?)1185 2800 y Fh(\026)p Fx(h)1228 2807 y Fp(k)1257 2800 y Fw(;)919 2967 y Fy(19)p eop %%Page: 20 20 20 19 bop -12 50 a Fy(with)19 b Fx(h)119 57 y Fp(k)160 50 y Fy(=)h Fx(q)237 57 y Fp(k)271 50 y Fv(\010)13 b Fx(q)341 34 y Fu(?)341 63 y Fp(k)371 50 y Fy(,)20 b Fw(k)g Fy(=)g(0)p Fw(;)8 b Fy(1.)30 b(No)o(w)19 b(using)h(the)f(notation)g(of) g(Remark)g(3.2)f(and)h(the)h(t)o(wisted)f(dualit)o(y)-12 107 y(results)14 b(in)h(the)f(generic)g(p)q(osition)h(and)f(the)g(ab)q (elian)h(situations)f(\(Sections)h(5)e(and)h(6,)f(resp)q(ectiv)o(ely\)) j(w)o(e)d(ha)o(v)o(e)-12 163 y(already)i(sho)o(wn)g(the)h(follo)o(wing) g(equalities:)574 265 y Fv(M)p Fy(\()p Fx(q)669 272 y Ft(0)689 265 y Fy(\))707 246 y Fu(0)759 265 y Fy(=)841 252 y FB(f)836 265 y Fw(Z)867 272 y Ft(0)895 265 y Fv(M)p Fy(\()p Fx(q)990 246 y Fu(?)990 276 y Ft(0)1019 265 y Fy(\))1048 252 y FB(f)1044 265 y Fw(Z)1075 272 y Ft(0)1095 234 y Fu(\003)1127 265 y Fv(\032)d(L)p Fy(\()p Fx(F)1252 272 y Ft(0)1271 265 y Fy(\))526 b(\(31\))574 340 y Fv(M)p Fy(\()p Fx(q)669 347 y Ft(1)689 340 y Fy(\))707 321 y Fu(0)759 340 y Fy(=)841 327 y FB(f)836 340 y Fw(Z)867 347 y Ft(1)895 340 y Fv(M)p Fy(\()p Fx(q)990 321 y Fu(?)990 351 y Ft(1)1019 340 y Fy(\))1048 327 y FB(f)1044 340 y Fw(Z)1075 347 y Ft(1)1095 309 y Fu(\003)1127 340 y Fv(\032)13 b(L)p Fy(\()p Fx(F)1252 347 y Ft(1)1271 340 y Fy(\))8 b Fw(;)505 b Fy(\(32\))-12 442 y(where)21 b Fx(F)153 449 y Fp(k)194 442 y Fy(are)f(the)h(F)l(o)q(c)o(k)f(spaces)h (giv)o(en)g(b)o(y)f Fw(P)838 449 y Fp(k)881 442 y Fy(:=)i Fw(P)6 b Fh(\026)p Fx(h)1029 449 y Fp(k)1070 442 y Fy(on)21 b(whic)o(h)g(the)g(F)l(o)q(c)o(k)f(represen)o(tations)g Fw(\031)1817 449 y Fp(k)1859 442 y Fy(of)-12 498 y(CAR\()p Fx(h)130 505 y Fp(k)151 498 y Fw(;)8 b Fy(\000)200 505 y Fp(k)221 498 y Fy(\),)13 b Fw(k)g Fy(=)g(0)p Fw(;)8 b Fy(1,)k(act.)539 485 y FB(f)535 498 y Fw(Z)566 505 y Ft(0)598 498 y Fy(and)689 485 y FB(f)684 498 y Fw(Z)715 505 y Ft(1)748 498 y Fy(are)h(the)g(t)o(wist)f(op)q(erators)g (de\014ned)i(in)g(\(7\))e(from)g(the)h(corresp)q(ond-)-12 555 y(ing)k(implemen)o(ters)g Fw(Z)376 562 y Ft(0)412 555 y Fy(and)g Fw(Z)533 562 y Ft(1)569 555 y Fy(of)f(the)g(ev)o(en{o)q (ddness)h(automorphisms)f(in)h(the)f(resp)q(ectiv)o(e)i(F)l(o)q(c)o(k)e (spaces.)-12 611 y(T)l(o)g(b)q(e)i(able)g(to)e(apply)h(the)g(preceding) i(equations)e(to)f(the)h(presen)o(t)g(most)f(general)h(case)g(w)o(e)f (will)j(write)e(the)-12 668 y(F)l(o)q(c)o(k)e(represen)o(tation)h Fw(\031)i Fy(of)d(CAR\()p Fx(h)p Fw(;)8 b Fy(\000\))15 b(asso)q(ciated)h(to)f(the)h(basis)g(pro)s(jection)f Fw(P)23 b Fy(o)o(v)o(er)15 b(the)g(space)h Fx(F)1772 675 y Ft(0)1802 668 y Fv(\012)11 b Fx(F)1876 675 y Ft(1)-12 724 y Fy(as)k(in)h(Prop)q(osition)f(2.4.)59 781 y(The)f(pro)q(of)f(of)h (our)f(main)h(theorem)g(later)g(on)g(will)h(b)q(e)g(based)f(on)g(the)g (computation)g(of)f(the)h(comm)o(utan)o(ts)-12 837 y(giv)o(en)i(in)g (the)f(follo)o(wing)h(t)o(w)o(o)e(lemmas.)-12 943 y Fq(7.1)i(Lemma)24 b Fs(With)17 b(the)g(pr)n(e)n(c)n(e)n(ding)e(notation)h(we)g(have)125 1009 y FB(n)153 1056 y Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)261 1063 y Ft(0)280 1056 y Fy(\)\))c Fv(j)g Fw(q)373 1063 y Ft(0)406 1056 y Fv(2)h Fx(q)471 1063 y Ft(0)491 1009 y FB(o)519 1020 y Fu(0)572 1056 y Fy(=)653 1043 y FB(f)649 1056 y Fw(Z)680 1063 y Ft(0)707 1056 y Fv(M)p Fy(\()p Fx(q)802 1037 y Fu(?)802 1067 y Ft(0)831 1056 y Fy(\))861 1043 y FB(f)856 1056 y Fw(Z)887 1063 y Ft(0)907 1025 y Fu(\003)p 947 1025 36 2 v 947 1056 a Fv(\012)20 b(L)p Fy(\()p Fx(F)1079 1063 y Ft(1)1099 1056 y Fy(\))572 1196 y(=)649 1124 y FB(\( )775 1168 y Fw(E)812 1149 y Ft(+)809 1180 y(0)840 1168 y Fw(M)5 b(E)926 1149 y Ft(+)923 1180 y(0)985 1168 y Fw(iE)1038 1149 y Ft(+)1035 1180 y(0)1066 1168 y Fw(M)g(E)1152 1149 y Fu(\000)1149 1180 y Ft(0)724 1224 y Fv(\000)p Fw(iE)812 1205 y Fu(\000)809 1236 y Ft(0)840 1224 y Fw(M)g(E)926 1205 y Ft(+)923 1236 y(0)985 1224 y Fw(E)1022 1205 y Fu(\000)1019 1236 y Ft(0)1051 1224 y Fw(M)g(E)1137 1205 y Fu(\000)1134 1236 y Ft(0)1190 1124 y FB(!)1235 1196 y Fv(j)13 b Fw(M)k Fv(2)c(M)p Fy(\()p Fx(q)1460 1177 y Fu(?)1460 1207 y Ft(0)1489 1196 y Fy(\))1507 1124 y FB(\))p 1568 1164 V 1568 1196 a Fv(\012)21 b(L)p Fy(\()p Fx(F)1701 1203 y Ft(1)1720 1196 y Fy(\))8 b Fw(:)-12 1332 y Fs(Pr)n(o)n(of:)31 b Fy(Using)21 b(Prop)q(osition)g(2.4)f(and)g (standard)h(relations)g(for)f(the)g(tensor)h(pro)q(duct)g(of)f(v)o(on)g (Neumann)-12 1389 y(algebras)15 b([25)o(,)g(Section)h(11.2])e(w)o(e)h (get)90 1457 y FB(n)117 1504 y Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)225 1511 y Ft(0)244 1504 y Fy(\)\))d Fv(j)h Fw(q)338 1511 y Ft(0)370 1504 y Fv(2)g Fx(q)435 1511 y Ft(0)455 1457 y FB(o)483 1468 y Fu(0)536 1504 y Fy(=)613 1457 y FB(n)640 1504 y Fw(\031)666 1511 y Ft(0)686 1504 y Fy(\()p Fw(a)p Fy(\()p Fw(q)766 1511 y Ft(0)785 1504 y Fy(\)\))d Fv(\012)g Fy(1)-10 b FF(l)18 b Fv(j)12 b Fw(q)964 1511 y Ft(0)996 1504 y Fv(2)h Fx(q)1061 1511 y Ft(0)1081 1457 y FB(o)1109 1468 y Fu(0)1133 1504 y Fy(=)1181 1457 y FB(n)1209 1504 y Fw(\031)1235 1511 y Ft(0)1255 1504 y Fy(\()p Fw(a)p Fy(\()p Fw(q)1335 1511 y Ft(0)1354 1504 y Fy(\)\))f Fv(j)g Fw(q)1447 1511 y Ft(0)1480 1504 y Fv(2)h Fx(q)1545 1511 y Ft(0)1565 1457 y FB(o)1592 1468 y Fu(0)p 1624 1473 V 1624 1504 a Fv(\012)21 b(L)p Fy(\()p Fx(F)1757 1511 y Ft(1)1776 1504 y Fy(\))536 1593 y(=)42 b Fv(M)p Fy(\()p Fx(q)708 1600 y Ft(0)727 1593 y Fy(\))745 1574 y Fu(0)p 777 1562 V 777 1593 a Fv(\012)20 b(L)p Fy(\()p Fx(F)909 1600 y Ft(1)929 1593 y Fy(\))12 b(=)1011 1580 y FB(f)1007 1593 y Fw(Z)1038 1600 y Ft(0)1065 1593 y Fv(M)p Fy(\()p Fx(q)1160 1574 y Fu(?)1160 1604 y Ft(0)1189 1593 y Fy(\))1219 1580 y FB(f)1214 1593 y Fw(Z)1245 1600 y Ft(0)1265 1562 y Fu(\003)p 1305 1562 V 1305 1593 a Fv(\012)21 b(L)p Fy(\()p Fx(F)1438 1600 y Ft(1)1457 1593 y Fy(\))8 b Fw(;)-12 1695 y Fy(where)15 b(for)g(the)g(last)g(equation)h(w)o(e)f(ha)o(v)o(e)g(used)g(Eq.)g (\(31\).)59 1752 y(Finally)l(,)g(the)g(last)f(equation)g(is)h(obtained) g(writing)964 1739 y FB(e)955 1752 y Fw(Z)986 1759 y Ft(0)1018 1752 y Fy(=)e Fw(E)1103 1733 y Ft(+)1100 1764 y(0)1140 1752 y Fv(\000)8 b Fw(iE)1236 1733 y Fu(\000)1233 1764 y Ft(0)1279 1752 y Fy(\(cf.)13 b(Eq.)h(\(8\)\))f(in)i(the)g (\014rst)e(term)h(of)-12 1808 y(the)h(preceding)i(tensor)e(pro)q(duct.) p 1857 1808 24 24 v 59 1900 a(F)l(or)20 b(the)h(calculation)i(of)e(the) g(other)g(comm)o(utan)o(t)f(needed,)k(tak)o(e)c(in)o(to)h(accoun)o(t)g (the)g(decomp)q(osition)-12 1957 y Fx(F)16 1964 y Ft(0)55 1957 y Fy(=)e Fx(F)137 1938 y Ft(+)137 1969 y(0)179 1957 y Fv(\010)13 b Fx(F)255 1938 y Fu(\000)255 1969 y Ft(0)284 1957 y Fy(,)20 b(where)f Fx(F)480 1938 y Fu(\006)480 1969 y Ft(0)529 1957 y Fy(:=)g Fw(E)633 1938 y Fu(\006)630 1969 y Ft(0)662 1957 y Fx(F)690 1964 y Ft(0)728 1957 y Fy(and)g Fw(Z)851 1964 y Ft(0)890 1957 y Fy(=)h Fw(E)982 1938 y Ft(+)979 1969 y(0)1023 1957 y Fv(\000)13 b Fw(E)1108 1938 y Fu(\000)1105 1969 y Ft(0)1156 1957 y Fy(\(cf.)18 b(Eq.)h(\(4\)\).)30 b(In)20 b(particular)g(an)o(y)f Fw(C)i Fv(2)-12 2013 y(L)p Fy(\()p Fx(F)65 2020 y Ft(0)95 2013 y Fv(\012)10 b Fx(F)168 2020 y Ft(1)188 2013 y Fy(\))218 2001 y Fv(\030)218 2015 y Fy(=)266 2013 y Fv(L)p Fy(\()p Fx(F)343 1994 y Ft(+)343 2025 y(0)382 2013 y Fv(\012)h Fx(F)456 2020 y Ft(1)485 2013 y Fv(\010)g Fx(F)559 1994 y Fu(\000)559 2025 y Ft(0)598 2013 y Fv(\012)g Fx(F)672 2020 y Ft(1)691 2013 y Fy(\))k(can)g(b)q(e)h(written)f(as)285 2077 y FB( )315 2121 y Fw(C)348 2128 y Ft(++)422 2121 y Fw(C)455 2128 y Ft(+)p Fu(\000)315 2177 y Fw(C)348 2184 y Fu(\000)p Ft(+)422 2177 y Fw(C)455 2184 y Fu(\000\000)509 2077 y FB(!)557 2149 y Fw(;)53 b Fy(where)13 b Fw(C)785 2156 y Fp(j)r(k)834 2149 y Fv(2)g(L)p Fy(\()p Fx(F)954 2130 y Fp(k)954 2160 y Ft(0)985 2149 y Fv(\012)e Fx(F)1059 2156 y Ft(1)1091 2149 y Fv(!)i Fx(F)1177 2127 y Fp(j)1177 2161 y Ft(0)1207 2149 y Fv(\012)d Fx(F)1280 2156 y Ft(1)1300 2149 y Fy(\))e Fw(;)19 b(j;)8 b(k)k Fy(=)h(+)p Fw(;)8 b Fv(\000)13 b Fw(:)-12 2285 y Fq(7.2)j(Lemma)24 b Fs(With)17 b(the)g(pr)n(e)n(c)n(e)n(ding)e(c)n(onventions)f(we)i(have)15 2419 y Fv(f)p Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)146 2426 y Ft(1)164 2419 y Fy(\)\))c Fv(j)g Fw(q)257 2426 y Ft(1)290 2419 y Fv(2)h Fx(q)355 2426 y Ft(1)375 2419 y Fv(g)398 2400 y Fu(0)422 2419 y Fy(=)470 2347 y FB( )604 2391 y Fv(L)p Fy(\()p Fx(F)681 2372 y Ft(+)681 2403 y(0)710 2391 y Fy(\))p 740 2359 36 2 v 12 w Fv(\012)793 2377 y FB(f)788 2391 y Fw(Z)819 2398 y Ft(1)839 2391 y Fv(M)p Fy(\()p Fx(q)934 2374 y Fu(?)934 2402 y Ft(1)963 2391 y Fy(\))986 2377 y FB(f)981 2391 y Fw(Z)1012 2398 y Ft(1)1031 2359 y Fu(\003)1173 2391 y Fv(L)p Fy(\()p Fx(F)1250 2372 y Fu(\000)1250 2403 y Ft(0)1284 2391 y Fv(!)5 b Fx(F)1362 2372 y Ft(+)1362 2403 y(0)1392 2391 y Fy(\))p 1422 2359 V 12 w Fv(\012)13 b Fy(\()p Fv(\000)p Fw(i)1544 2377 y FB(f)1539 2391 y Fw(Z)1570 2398 y Ft(1)1589 2391 y Fv(M)p Fy(\()p Fx(q)1684 2374 y Fu(?)1684 2402 y Ft(1)1713 2391 y Fy(\))1736 2377 y FB(f)1731 2391 y Fw(Z)1762 2398 y Ft(1)1782 2391 y Fy(\))512 2454 y Fv(L)p Fy(\()p Fx(F)589 2435 y Ft(+)589 2467 y(0)623 2454 y Fv(!)5 b Fx(F)701 2435 y Fu(\000)701 2467 y Ft(0)731 2454 y Fy(\))p 761 2423 V 12 w Fv(\012)13 b Fy(\()p Fw(i)848 2441 y FB(f)843 2454 y Fw(Z)874 2461 y Ft(1)893 2423 y Fu(\003)913 2454 y Fv(M)p Fy(\()p Fx(q)1008 2438 y Fu(?)1008 2466 y Ft(1)1037 2454 y Fy(\))1060 2441 y FB(f)1055 2454 y Fw(Z)1086 2461 y Ft(1)1105 2423 y Fu(\003)1125 2454 y Fy(\))119 b Fv(L)p Fy(\()p Fx(F)1339 2435 y Fu(\000)1339 2467 y Ft(0)1369 2454 y Fy(\))p 1399 2423 V 12 w Fv(\012)1452 2441 y FB(f)1447 2454 y Fw(Z)1478 2461 y Ft(1)1498 2454 y Fv(M)p Fy(\()p Fx(q)1593 2438 y Fu(?)1593 2466 y Ft(1)1622 2454 y Fy(\))1645 2441 y FB(f)1640 2454 y Fw(Z)1671 2461 y Ft(1)1690 2423 y Fu(\003)1808 2347 y FB(!)1857 2419 y Fw(:)-12 2559 y Fs(Pr)n(o)n(of:)20 b Fy(First)15 b(note)g(that)f(from)h(Corollary)g (2.5,)f Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)941 2566 y Ft(1)959 2559 y Fy(\)\))h(can)g(b)q(e)h(written)f(as)g(a)g(diagonal)h (op)q(erator)347 2694 y Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)455 2701 y Ft(1)474 2694 y Fy(\)\))c(=)570 2622 y FB( )600 2666 y Fy(1)-11 b FF(l)629 2673 y Ft(+)669 2666 y Fv(\012)10 b Fw(\031)740 2673 y Ft(1)760 2666 y Fy(\()p Fw(a)p Fy(\()p Fw(q)840 2673 y Ft(1)859 2666 y Fy(\)\))204 b(0)736 2722 y(0)153 b(1)-10 b FF(l)942 2729 y Fu(\000)982 2722 y Fv(\012)10 b Fw(Z)1058 2729 y Ft(1)1078 2722 y Fw(\031)1104 2729 y Ft(1)1123 2722 y Fy(\()p Fw(a)p Fy(\()p Fw(q)1203 2729 y Ft(1)1223 2722 y Fy(\)\))p Fw(Z)1290 2729 y Ft(1)1306 2622 y FB(!)1359 2694 y Fw(;)e(q)1400 2701 y Ft(1)1432 2694 y Fv(2)13 b Fx(q)1497 2701 y Ft(1)1524 2694 y Fw(;)919 2967 y Fy(20)p eop %%Page: 21 21 21 20 bop -12 50 a Fy(where)17 b(1)-10 b FF(l)151 57 y Fu(\006)197 50 y Fy(are)17 b(the)g(unit)h(op)q(erators)e(on)h Fx(F)749 31 y Fu(\006)749 63 y Ft(0)779 50 y Fy(.)25 b(Therefore,)17 b(with)h(the)f(notation)f(ab)q(o)o(v)o(e)h(w)o(e)g(ha)o (v)o(e)g(that)f Fw(C)i Fv(2)-12 107 y(f)p Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)119 114 y Ft(1)138 107 y Fy(\)\))12 b Fv(j)g Fw(q)231 114 y Ft(1)263 107 y Fv(2)h Fx(q)328 114 y Ft(1)348 107 y Fv(g)371 90 y Fu(0)398 107 y Fy(i\013)56 155 y FB( )86 198 y Fw(C)119 205 y Ft(++)193 198 y Fw(C)226 205 y Ft(+)p Fu(\000)86 255 y Fw(C)119 262 y Fu(\000)p Ft(+)193 255 y Fw(C)226 262 y Fu(\000\000)280 155 y FB(!)323 227 y Fv(\001)345 155 y FB( )375 198 y Fy(1)-10 b FF(l)405 205 y Ft(+)445 198 y Fv(\012)10 b Fw(\031)516 205 y Ft(1)536 198 y Fy(\()p Fw(a)p Fy(\()p Fw(q)616 205 y Ft(1)635 198 y Fy(\)\))204 b(0)512 255 y(0)153 b(1)-10 b FF(l)718 262 y Fu(\000)758 255 y Fv(\012)10 b Fw(Z)834 262 y Ft(1)854 255 y Fw(\031)880 262 y Ft(1)899 255 y Fy(\()p Fw(a)p Fy(\()p Fw(q)979 262 y Ft(1)999 255 y Fy(\)\))p Fw(Z)1066 262 y Ft(1)1082 155 y FB(!)279 368 y Fy(=)356 296 y FB( )386 340 y Fy(1)-11 b FF(l)416 347 y Ft(+)455 340 y Fv(\012)11 b Fw(\031)527 347 y Ft(1)546 340 y Fy(\()p Fw(a)p Fy(\()p Fw(q)626 347 y Ft(1)646 340 y Fy(\)\))204 b(0)522 396 y(0)154 b(1)-11 b FF(l)728 403 y Fu(\000)768 396 y Fv(\012)11 b Fw(Z)845 403 y Ft(1)864 396 y Fw(\031)890 403 y Ft(1)910 396 y Fy(\()p Fw(a)p Fy(\()p Fw(q)990 403 y Ft(1)1009 396 y Fy(\)\))p Fw(Z)1076 403 y Ft(1)1093 296 y FB(!)1136 368 y Fv(\001)1158 296 y FB( )1188 340 y Fw(C)1221 347 y Ft(++)1296 340 y Fw(C)1329 347 y Ft(+)p Fu(\000)1188 396 y Fw(C)1221 403 y Fu(\000)p Ft(+)1296 396 y Fw(C)1329 403 y Fu(\000\000)1382 296 y FB(!)1423 368 y Fw(;)52 b(q)1508 375 y Ft(1)1541 368 y Fv(2)13 b Fx(q)1606 375 y Ft(1)1633 368 y Fw(:)-12 486 y Fy(W)l(e)i(can)h(no)o(w)e(rewrite)i (the)f(preceding)i(equation)e(as)347 573 y Fw(C)380 580 y Ft(++)437 573 y Fy(\(1)-11 b FF(l)484 580 y Ft(+)524 573 y Fv(\012)10 b Fw(\031)595 580 y Ft(1)615 573 y Fy(\()p Fw(a)p Fy(\()p Fw(q)695 580 y Ft(1)714 573 y Fy(\)\)\))41 b(=)h(\(1)-11 b FF(l)933 580 y Ft(+)972 573 y Fv(\012)11 b Fw(\031)1044 580 y Ft(1)1063 573 y Fy(\()p Fw(a)p Fy(\()p Fw(q)1143 580 y Ft(1)1163 573 y Fy(\)\)\))p Fw(C)1250 580 y Ft(++)1313 573 y Fw(;)53 b(q)1399 580 y Ft(1)1431 573 y Fv(2)13 b Fx(q)1496 580 y Ft(1)1524 573 y Fw(;)278 b Fy(\(33\))347 665 y Fw(C)380 672 y Fu(\000\000)437 665 y Fy(\(1)-11 b FF(l)484 672 y Fu(\000)524 665 y Fv(\012)10 b Fw(\031)595 672 y Ft(1)615 665 y Fy(\()p Fw(a)p Fy(\()p Fw(q)695 672 y Ft(1)714 665 y Fy(\)\)\))41 b(=)h(\(1)-11 b FF(l)933 672 y Fu(\000)972 665 y Fv(\012)11 b Fw(\031)1044 672 y Ft(1)1063 665 y Fy(\()p Fw(a)p Fy(\()p Fw(q)1143 672 y Ft(1)1163 665 y Fy(\)\)\))p Fw(C)1250 672 y Fu(\000\000)1313 665 y Fw(;)53 b(q)1399 672 y Ft(1)1431 665 y Fv(2)13 b Fx(q)1496 672 y Ft(1)1524 665 y Fw(;)278 b Fy(\(34\))246 758 y Fw(C)279 765 y Ft(+)p Fu(\000)335 758 y Fy(\(1)-11 b FF(l)383 765 y Fu(\000)422 758 y Fv(\012)11 b Fw(Z)499 765 y Ft(1)518 758 y Fw(\031)544 765 y Ft(1)564 758 y Fy(\()p Fw(a)p Fy(\()p Fw(q)644 765 y Ft(1)663 758 y Fy(\)\))p Fw(Z)730 765 y Ft(1)749 758 y Fy(\))42 b(=)g(\(1)-11 b FF(l)933 765 y Ft(+)972 758 y Fv(\012)11 b Fw(\031)1044 765 y Ft(1)1063 758 y Fy(\()p Fw(a)p Fy(\()p Fw(q)1143 765 y Ft(1)1163 758 y Fy(\)\)\))p Fw(C)1250 765 y Ft(+)p Fu(\000)1313 758 y Fw(;)53 b(q)1399 765 y Ft(1)1431 758 y Fv(2)13 b Fx(q)1496 765 y Ft(1)1524 758 y Fw(;)278 b Fy(\(35\))347 851 y Fw(C)380 858 y Fu(\000)p Ft(+)437 851 y Fy(\(1)-11 b FF(l)484 858 y Ft(+)524 851 y Fv(\012)10 b Fw(\031)595 858 y Ft(1)615 851 y Fy(\()p Fw(a)p Fy(\()p Fw(q)695 858 y Ft(1)714 851 y Fy(\)\)\))41 b(=)h(\(1)-11 b FF(l)933 858 y Fu(\000)972 851 y Fv(\012)11 b Fw(Z)1049 858 y Ft(1)1069 851 y Fw(\031)1095 858 y Ft(1)1114 851 y Fy(\()p Fw(a)p Fy(\()p Fw(q)1194 858 y Ft(1)1214 851 y Fy(\)\))p Fw(Z)1281 858 y Ft(1)1300 851 y Fy(\))p Fw(C)1351 858 y Fu(\000)p Ft(+)1414 851 y Fw(;)53 b(q)1500 858 y Ft(1)1533 851 y Fv(2)13 b Fx(q)1598 858 y Ft(1)1625 851 y Fw(:)177 b Fy(\(36\))-12 938 y(Eqs.)15 b(\(33\))f(and)h(\(34\))f (imply)i(simply)h(that)108 1039 y Fw(C)141 1046 y Ft(++)224 1039 y Fv(2)280 991 y FB(\020)305 1039 y Fv(f)p Fy(1)-11 b FF(l)357 1046 y Ft(+)387 1039 y Fv(g)10 b(\012)g(f)p Fw(\031)514 1046 y Ft(1)533 1039 y Fy(\()p Fw(a)p Fy(\()p Fw(q)613 1046 y Ft(1)633 1039 y Fy(\)\))p Fv(g)692 991 y FB(\021)716 1003 y Fu(0)740 1039 y Fy(=)j Fv(f)p Fy(1)-11 b FF(l)840 1046 y Ft(+)870 1039 y Fv(g)893 1020 y Fu(0)p 924 1007 36 2 v 924 1039 a Fv(\012)21 b(f)p Fw(\031)1029 1046 y Ft(1)1048 1039 y Fy(\()p Fw(a)p Fy(\()p Fw(q)1128 1046 y Ft(1)1148 1039 y Fy(\)\))p Fv(g)1207 1020 y Fu(0)1230 1039 y Fy(=)13 b Fv(L)p Fy(\()p Fx(F)1355 1019 y Ft(+)1355 1051 y(0)1384 1039 y Fy(\))p 1422 1007 V 20 w Fv(\012)1482 1025 y FB(f)1478 1039 y Fw(Z)1509 1046 y Ft(1)1536 1039 y Fv(M)p Fy(\()p Fx(q)1631 1020 y Fu(?)1631 1050 y Ft(1)1660 1039 y Fy(\))1690 1025 y FB(f)1685 1039 y Fw(Z)1716 1046 y Ft(1)1736 1007 y Fu(\003)108 1135 y Fw(C)141 1142 y Fu(\000\000)224 1135 y Fv(2)280 1088 y FB(\020)305 1135 y Fv(f)p Fy(1)-11 b FF(l)357 1142 y Fu(\000)387 1135 y Fv(g)10 b(\012)g(f)p Fw(\031)514 1142 y Ft(1)533 1135 y Fy(\()p Fw(a)p Fy(\()p Fw(q)613 1142 y Ft(1)633 1135 y Fy(\)\))p Fv(g)692 1088 y FB(\021)716 1100 y Fu(0)740 1135 y Fy(=)j Fv(f)p Fy(1)-11 b FF(l)840 1142 y Fu(\000)870 1135 y Fv(g)893 1116 y Fu(0)p 924 1104 V 924 1135 a Fv(\012)21 b(f)p Fw(\031)1029 1142 y Ft(1)1048 1135 y Fy(\()p Fw(a)p Fy(\()p Fw(q)1128 1142 y Ft(1)1148 1135 y Fy(\)\))p Fv(g)1207 1116 y Fu(0)1230 1135 y Fy(=)13 b Fv(L)p Fy(\()p Fx(F)1355 1116 y Fu(\000)1355 1148 y Ft(0)1384 1135 y Fy(\))p 1422 1104 V 20 w Fv(\012)1482 1122 y FB(f)1478 1135 y Fw(Z)1509 1142 y Ft(1)1536 1135 y Fv(M)p Fy(\()p Fx(q)1631 1116 y Fu(?)1631 1146 y Ft(1)1660 1135 y Fy(\))1690 1122 y FB(f)1685 1135 y Fw(Z)1716 1142 y Ft(1)1736 1104 y Fu(\003)1763 1135 y Fw(;)-12 1235 y Fy(where)18 b(w)o(e)g(ha)o(v)o(e)f(used)i(Eq.)e (\(32\).)27 b(T)l(o)17 b(obtain)h Fw(C)859 1242 y Ft(+)p Fu(\000)934 1235 y Fy(w)o(e)f(need)i(an)f(isometry)f Fw(K)s Fy(:)8 b Fx(F)1457 1216 y Ft(+)1457 1248 y(0)1503 1235 y Fv(!)17 b Fx(F)1593 1216 y Fu(\000)1593 1248 y Ft(0)1623 1235 y Fy(.)27 b(Multiplying)-12 1292 y(Eq.)15 b(\(35\))f(from)g(the)h(righ)o(t)g(b)o(y)g(\()p Fw(K)e Fv(\012)e Fw(Z)672 1299 y Ft(1)691 1292 y Fy(\))k(and)g(using)h(1)-10 b FF(l)961 1299 y Fu(\000)990 1292 y Fw(K)16 b Fy(=)d Fw(K)s Fy(1)-11 b FF(l)1164 1299 y Ft(+)1209 1292 y Fy(w)o(e)15 b(obtain)371 1379 y Fw(D)409 1386 y Ft(+)p Fu(\000)466 1379 y Fy(\(1)-11 b FF(l)513 1386 y Ft(+)553 1379 y Fv(\012)10 b Fw(\031)624 1386 y Ft(1)644 1379 y Fy(\()p Fw(a)p Fy(\()p Fw(q)724 1386 y Ft(1)743 1379 y Fy(\)\)\))i(=)h(\(1)-11 b FF(l)904 1386 y Ft(+)943 1379 y Fv(\012)11 b Fw(\031)1015 1386 y Ft(1)1035 1379 y Fy(\()p Fw(a)p Fy(\()p Fw(q)1115 1386 y Ft(1)1134 1379 y Fy(\)\)\))p Fw(D)1226 1386 y Ft(+)p Fu(\000)1289 1379 y Fw(;)53 b(q)1375 1386 y Ft(1)1407 1379 y Fv(2)13 b Fx(q)1472 1386 y Ft(1)1500 1379 y Fw(;)-12 1467 y Fy(where)27 b Fw(D)169 1474 y Ft(+)p Fu(\000)257 1467 y Fy(:=)k Fw(C)369 1474 y Ft(+)p Fu(\000)425 1467 y Fy(\()p Fw(K)20 b Fv(\012)e Fw(Z)586 1474 y Ft(1)606 1467 y Fy(\))31 b Fv(2)h(L)p Fy(\()p Fx(F)794 1448 y Ft(+)794 1479 y(0)823 1467 y Fy(\))p 880 1435 V 39 w Fv(\012)39 b(L)p Fy(\()p Fx(F)1031 1474 y Ft(1)1050 1467 y Fy(\).)54 b(This)27 b(implies)h(as)e(b)q(efore)h(that)e Fw(D)1777 1474 y Ft(+)p Fu(\000)1865 1467 y Fv(2)-12 1530 y(L)p Fy(\()p Fx(F)65 1511 y Ft(+)65 1542 y(0)94 1530 y Fy(\))p 134 1498 V 22 w Fv(\012)196 1517 y FB(f)191 1530 y Fw(Z)222 1537 y Ft(1)249 1530 y Fv(M)p Fy(\()p Fx(q)344 1513 y Fu(?)344 1542 y Ft(1)373 1530 y Fy(\))403 1517 y FB(f)399 1530 y Fw(Z)430 1537 y Ft(1)449 1499 y Fu(\003)469 1530 y Fy(.)e(No)o(w)16 b(from)f(the)h(de\014nition)i(of) e Fw(D)1091 1537 y Ft(+)p Fu(\000)1164 1530 y Fy(and)g(from)g(the)g (relation)1613 1517 y FB(f)1609 1530 y Fw(Z)1640 1537 y Ft(1)1660 1499 y Fu(\003)1679 1530 y Fw(Z)1710 1537 y Ft(1)1744 1530 y Fy(=)f Fv(\000)p Fw(i)1850 1517 y FB(f)1845 1530 y Fw(Z)1876 1537 y Ft(1)-12 1586 y Fy(w)o(e)g(get)g (\014nally)531 1643 y Fw(C)564 1650 y Ft(+)p Fu(\000)633 1643 y Fv(2)e(L)p Fy(\()p Fx(F)753 1624 y Fu(\000)753 1655 y Ft(0)795 1643 y Fv(!)g Fx(F)881 1624 y Ft(+)881 1655 y(0)910 1643 y Fy(\))p 948 1611 V 20 w Fv(\012)20 b Fy(\()p Fv(\000)p Fw(i)1077 1630 y FB(f)1072 1643 y Fw(Z)1103 1650 y Ft(1)1123 1643 y Fv(M)p Fy(\()p Fx(q)1218 1624 y Fu(?)1218 1654 y Ft(1)1247 1643 y Fy(\))1270 1630 y FB(f)1265 1643 y Fw(Z)1296 1650 y Ft(1)1315 1643 y Fy(\))8 b Fw(:)-12 1719 y Fy(In)16 b(a)f(similar)h(w)o(a)o(y)e(w)o(e)h (obtain)h(the)f(corresp)q(onding)h(relation)g(for)e Fw(C)1172 1726 y Fu(\000)p Ft(+)1229 1719 y Fy(.)p 1857 1719 24 24 v -12 1846 a Fq(7.3)i(Theo)o(rem)23 b Fs(\(Twiste)n(d)16 b(Duality\))h(L)n(et)e Fy(\()p Fx(h)p Fw(;)8 b Fy(\000\))p Fs(,)15 b Fw(P)23 b Fs(and)17 b Fx(q)g Fs(b)n(e)f(given)g(as)g(in)g (the)h(b)n(e)n(ginning)d(of)i(this)h(se)n(ction.)-12 1902 y(Then)717 1959 y Fv(M)p Fy(\()p Fx(q)p Fy(\))830 1940 y Fu(0)853 1959 y Fy(=)911 1946 y FB(e)901 1959 y Fw(Z)11 b Fv(M)p Fy(\()p Fx(q)1038 1940 y Fu(?)1067 1959 y Fy(\))1102 1946 y FB(e)1093 1959 y Fw(Z)1127 1940 y Fu(\003)1154 1959 y Fw(:)-12 2050 y Fs(Pr)n(o)n(of:)20 b Fy(F)l(rom)14 b(Prop)q(osition)i(2.6)e(it)i(is)f(enough)h(to)e(sho)o (w)h(the)g(inclusion)717 2138 y Fv(M)p Fy(\()p Fx(q)p Fy(\))830 2119 y Fu(0)853 2138 y Fv(\022)911 2125 y FB(e)901 2138 y Fw(Z)c Fv(M)p Fy(\()p Fx(q)1038 2119 y Fu(?)1067 2138 y Fy(\))1102 2125 y FB(e)1093 2138 y Fw(Z)1127 2119 y Fu(\003)1154 2138 y Fw(:)648 b Fy(\(37\))-12 2225 y(No)o(w,)14 b(for)h(the)g(l.h.s.)g(of)g(the)g(preceding)i(inclusion)g(w)o(e)e(ha)o (v)o(e)117 2313 y Fv(M)p Fy(\()p Fx(q)p Fy(\))230 2294 y Fu(0)283 2313 y Fy(=)42 b Fv(f)p Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)491 2320 y Ft(0)519 2313 y Fy(+)11 b Fw(q)585 2320 y Ft(1)605 2313 y Fy(\)\))h Fv(j)g Fw(q)698 2320 y Ft(0)728 2313 y Fy(+)f Fw(q)794 2320 y Ft(1)826 2313 y Fv(2)i Fx(q)g Fy(=)g Fx(q)974 2320 y Ft(0)1004 2313 y Fv(\010)d Fx(q)1071 2320 y Ft(1)1091 2313 y Fv(g)1114 2294 y Fu(0)283 2418 y Fy(=)360 2371 y FB(\020)384 2418 y Fv(f)p Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)515 2425 y Ft(0)534 2418 y Fy(\)\))i Fv(j)g Fw(q)627 2425 y Ft(0)660 2418 y Fv(2)h Fx(q)725 2425 y Ft(0)745 2418 y Fv(g)d([)g(f)p Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)949 2425 y Ft(1)968 2418 y Fy(\)\))i Fv(j)g Fw(q)1061 2425 y Ft(1)1093 2418 y Fv(2)h Fx(q)1158 2425 y Ft(1)1178 2418 y Fv(g)1201 2371 y FB(\021)1226 2383 y Fu(0)283 2522 y Fy(=)42 b Fv(f)p Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)491 2529 y Ft(0)509 2522 y Fy(\)\))12 b Fv(j)g Fw(q)602 2529 y Ft(0)635 2522 y Fv(2)h Fx(q)700 2529 y Ft(0)720 2522 y Fv(g)743 2503 y Fu(0)764 2522 y Fv(\\)e(f)p Fw(\031)r Fy(\()p Fw(a)p Fy(\()p Fw(q)936 2529 y Ft(1)954 2522 y Fy(\)\))h Fv(j)h Fw(q)1048 2529 y Ft(1)1080 2522 y Fv(2)g Fx(q)1145 2529 y Ft(1)1165 2522 y Fv(g)1188 2503 y Fu(0)283 2647 y Fy(=)360 2575 y FB( )o(\()q( )519 2619 y Fw(E)556 2600 y Ft(+)553 2631 y(0)584 2619 y Fw(M)5 b(E)670 2600 y Ft(+)667 2631 y(0)729 2619 y Fw(iE)782 2600 y Ft(+)779 2631 y(0)810 2619 y Fw(M)g(E)896 2600 y Fu(\000)893 2631 y Ft(0)468 2675 y Fv(\000)p Fw(iE)556 2656 y Fu(\000)553 2688 y Ft(0)584 2675 y Fw(M)g(E)670 2656 y Ft(+)667 2688 y(0)729 2675 y Fw(E)766 2656 y Fu(\000)763 2688 y Ft(0)794 2675 y Fw(M)g(E)880 2656 y Fu(\000)877 2688 y Ft(0)934 2575 y FB(!)979 2647 y Fv(j)13 b Fw(M)k Fv(2)c(M)p Fy(\()p Fx(q)1204 2628 y Fu(?)1204 2658 y Ft(0)1233 2647 y Fy(\))1251 2575 y FB(\))p 1312 2616 36 2 v 1312 2647 a Fv(\012)21 b(L)p Fy(\()p Fx(F)1445 2654 y Ft(1)1464 2647 y Fy(\))1482 2575 y FB(!)329 2813 y Fv(\\)367 2741 y FB( )501 2785 y Fv(L)p Fy(\()p Fx(F)578 2766 y Ft(+)578 2797 y(0)607 2785 y Fy(\))p 637 2753 V 12 w Fv(\012)690 2772 y FB(f)685 2785 y Fw(Z)716 2792 y Ft(1)736 2785 y Fv(M)p Fy(\()p Fx(q)831 2768 y Fu(?)831 2796 y Ft(1)860 2785 y Fy(\))883 2772 y FB(f)878 2785 y Fw(Z)909 2792 y Ft(1)929 2754 y Fu(\003)1070 2785 y Fv(L)p Fy(\()p Fx(F)1147 2766 y Fu(\000)1147 2797 y Ft(0)1181 2785 y Fv(!)5 b Fx(F)1259 2766 y Ft(+)1259 2797 y(0)1289 2785 y Fy(\))p 1319 2753 V 12 w Fv(\012)13 b Fy(\()p Fv(\000)p Fw(i)1441 2772 y FB(f)1436 2785 y Fw(Z)1467 2792 y Ft(1)1487 2785 y Fv(M)p Fy(\()p Fx(q)1582 2768 y Fu(?)1582 2796 y Ft(1)1611 2785 y Fy(\))1634 2772 y FB(f)1629 2785 y Fw(Z)1660 2792 y Ft(1)1679 2785 y Fy(\))409 2848 y Fv(L)p Fy(\()p Fx(F)486 2829 y Ft(+)486 2861 y(0)520 2848 y Fv(!)5 b Fx(F)598 2829 y Fu(\000)598 2861 y Ft(0)628 2848 y Fy(\))p 658 2817 V 12 w Fv(\012)13 b Fy(\()p Fw(i)745 2835 y FB(f)740 2848 y Fw(Z)771 2855 y Ft(1)790 2817 y Fu(\003)810 2848 y Fv(M)p Fy(\()p Fx(q)905 2832 y Fu(?)905 2860 y Ft(1)934 2848 y Fy(\))957 2835 y FB(f)952 2848 y Fw(Z)983 2855 y Ft(1)1003 2817 y Fu(\003)1022 2848 y Fy(\))119 b Fv(L)p Fy(\()p Fx(F)1236 2829 y Fu(\000)1236 2861 y Ft(0)1266 2848 y Fy(\))p 1296 2817 V 12 w Fv(\012)1349 2835 y FB(f)1344 2848 y Fw(Z)1375 2855 y Ft(1)1395 2848 y Fv(M)p Fy(\()p Fx(q)1490 2832 y Fu(?)1490 2860 y Ft(1)1519 2848 y Fy(\))1542 2835 y FB(f)1537 2848 y Fw(Z)1568 2855 y Ft(1)1587 2817 y Fu(\003)1706 2741 y FB(!)1754 2813 y Fw(;)919 2967 y Fy(21)p eop %%Page: 22 22 22 21 bop -12 50 a Fy(where)14 b(for)f(the)h(last)g(equation)g(w)o(e)g (ha)o(v)o(e)f(used)h(Lemmas)g(7.1)f(and)h(7.2.)19 b(No)o(w)13 b(applying)i([32)o(,)f(Corollary)f(5.10])-12 107 y(to)h(the)i (preceding)g(in)o(tersection)h(of)d(tensor)h(pro)q(ducts)h(w)o(e)e (obtain)i(\014nally)214 235 y Fv(M)p Fy(\()p Fx(q)p Fy(\))327 216 y Fu(0)379 235 y Fy(=)456 163 y FB( \( )615 206 y Fw(E)652 187 y Ft(+)649 219 y(0)681 206 y Fw(M)5 b(E)767 187 y Ft(+)764 219 y(0)825 206 y Fw(iE)878 187 y Ft(+)875 219 y(0)907 206 y Fw(M)g(E)993 187 y Fu(\000)990 219 y Ft(0)564 263 y Fv(\000)p Fw(iE)652 244 y Fu(\000)649 275 y Ft(0)681 263 y Fw(M)g(E)767 244 y Ft(+)764 275 y(0)825 263 y Fw(E)862 244 y Fu(\000)859 275 y Ft(0)891 263 y Fw(M)g(E)977 244 y Fu(\000)974 275 y Ft(0)1030 163 y FB(!)1076 235 y Fv(j)12 b Fw(M)18 b Fv(2)13 b(M)p Fy(\()p Fx(q)1301 216 y Fu(?)1301 246 y Ft(0)1330 235 y Fy(\))1348 163 y FB(\))1391 235 y Fv(\\)d(L)p Fy(\()p Fx(F)1508 242 y Ft(0)1528 235 y Fy(\))1546 163 y FB(!)p 453 346 36 2 v 453 377 a Fv(\012)517 305 y FB( )549 377 y Fv(L)p Fy(\()p Fx(F)626 384 y Ft(1)646 377 y Fy(\))g Fv(\\)714 305 y FB( )796 336 y(f)791 349 y Fw(Z)822 356 y Ft(1)842 349 y Fv(M)p Fy(\()p Fx(q)937 332 y Fu(?)937 360 y Ft(1)966 349 y Fy(\))989 336 y FB(f)984 349 y Fw(Z)1015 356 y Ft(1)1035 318 y Fu(\003)1119 349 y Fy(\()p Fv(\000)p Fw(i)1193 336 y FB(f)1188 349 y Fw(Z)1219 356 y Ft(1)1239 349 y Fv(M)p Fy(\()p Fx(q)1334 332 y Fu(?)1334 360 y Ft(1)1363 349 y Fy(\))1386 336 y FB(f)1381 349 y Fw(Z)1412 356 y Ft(1)1431 349 y Fy(\))756 412 y(\()p Fw(i)795 399 y FB(f)790 412 y Fw(Z)821 419 y Ft(1)840 381 y Fu(\003)860 412 y Fv(M)p Fy(\()p Fx(q)955 396 y Fu(?)955 424 y Ft(1)984 412 y Fy(\))1007 399 y FB(f)1002 412 y Fw(Z)1033 419 y Ft(1)1052 381 y Fu(\003)1072 412 y Fy(\))1157 399 y FB(f)1153 412 y Fw(Z)1184 419 y Ft(1)1203 412 y Fv(M)p Fy(\()p Fx(q)1298 396 y Fu(?)1298 424 y Ft(1)1327 412 y Fy(\))1350 399 y FB(f)1345 412 y Fw(Z)1376 419 y Ft(1)1396 381 y Fu(\003)1458 305 y FB(!!)379 555 y Fy(=)42 b(clo)512 562 y Fp(w)547 483 y FB(\()q( )674 527 y Fw(E)711 508 y Ft(+)708 539 y(0)739 527 y Fw(M)5 b(E)825 508 y Ft(+)822 539 y(0)864 527 y Fv(\012)11 b Fw(N)947 534 y Ft(1)996 527 y Fw(iE)1049 508 y Ft(+)1046 539 y(0)1077 527 y Fw(M)5 b(E)1163 508 y Fu(\000)1160 539 y Ft(0)1202 527 y Fv(\012)10 b Fw(N)1284 534 y Ft(2)623 583 y Fv(\000)p Fw(iE)711 564 y Fu(\000)708 595 y Ft(0)739 583 y Fw(M)5 b(E)825 564 y Ft(+)822 595 y(0)864 583 y Fv(\012)11 b Fw(N)947 590 y Ft(3)996 583 y Fw(E)1033 564 y Fu(\000)1030 595 y Ft(0)1061 583 y Fw(M)5 b(E)1147 564 y Fu(\000)1144 595 y Ft(0)1186 583 y Fv(\012)11 b Fw(N)1269 590 y Ft(4)1312 483 y FB(!)1358 555 y Fv(j)h Fw(M)18 b Fv(2)13 b(M)p Fy(\()p Fx(q)1583 536 y Fu(?)1583 566 y Ft(0)1612 555 y Fy(\))8 b Fw(;)164 b Fy(\(38\))597 671 y Fw(N)634 678 y Ft(1)654 671 y Fw(;)8 b(N)712 678 y Ft(4)743 671 y Fv(2)790 658 y FB(f)786 671 y Fw(Z)817 678 y Ft(1)836 671 y Fv(M)p Fy(\()p Fx(q)931 652 y Fu(?)931 682 y Ft(1)960 671 y Fy(\))983 658 y FB(f)978 671 y Fw(Z)1009 678 y Ft(1)1029 640 y Fu(\003)1056 671 y Fw(;)20 b(N)1126 678 y Ft(2)1145 671 y Fw(;)8 b(N)1208 652 y Fu(\003)1203 682 y Ft(3)1239 671 y Fv(2)13 b(\000)p Fw(i)1338 658 y FB(f)1333 671 y Fw(Z)1364 678 y Ft(1)1384 671 y Fv(M)p Fy(\()p Fx(q)1479 652 y Fu(?)1479 682 y Ft(1)1508 671 y Fy(\))1531 658 y FB(f)1526 671 y Fw(Z)1557 678 y Ft(1)1576 624 y FB(o)1619 671 y Fw(:)-12 774 y Fy(T)l(o)20 b(analyize)h(the)f (\014rst)g(factors)f(of)h(the)g(previous)g(tensor)g(pro)q(ducts)g (recall)i(Lemma)e(7.1)f(and)h(that)f(from)-12 831 y(Remark)c(2.8)f(w)o (e)h(ha)o(v)o(e)130 915 y FB(f)125 928 y Fw(Z)156 935 y Ft(0)183 928 y Fv(M)p Fy(\()p Fx(q)278 910 y Fu(?)278 940 y Ft(0)307 928 y Fy(\))337 915 y FB(f)333 928 y Fw(Z)364 935 y Ft(0)383 897 y Fu(\003)416 928 y Fy(=)464 881 y FB(n)491 928 y Fw(\031)r Fy(\()p Fw(Y)563 935 y Ft(0)p Fp(;)p Fk(ev)o(en)652 928 y Fy(\))9 b(+)i Fw(iZ)772 935 y Ft(0)791 928 y Fw(\031)r Fy(\()p Fw(Y)863 935 y Ft(0)p Fp(;)p Fk(o)q(dd)942 928 y Fy(\)\))h Fv(j)g Fw(Y)1041 935 y Ft(0)1074 928 y Fy(=)h Fw(Y)1148 935 y Ft(0)p Fp(;)p Fk(ev)o(en)1247 928 y Fy(+)e Fw(Y)1319 935 y Ft(0)p Fp(;)p Fk(o)q(dd)1411 928 y Fv(2)i Fy(CAR\()p Fx(q)1594 910 y Fu(?)1594 940 y Ft(0)1624 928 y Fw(;)8 b Fy(\000)1673 935 y Ft(0)1692 928 y Fy(\))1710 881 y FB(o)1737 893 y Fu(00)-12 1031 y Fy(as)15 b(w)o(ell)h(as)f(that)f(Lemma)h(2.3)g (implies)363 1129 y Fw(E)400 1110 y Ft(+)397 1142 y(0)429 1129 y Fw(\031)455 1136 y Ft(0)474 1129 y Fy(\()p Fw(Y)518 1136 y Ft(0)p Fp(;)p Fk(ev)o(en)607 1129 y Fy(\))p Fw(E)662 1110 y Fu(\000)659 1142 y Ft(0)732 1129 y Fy(=)42 b Fw(E)846 1110 y Fu(\000)843 1142 y Ft(0)875 1129 y Fw(\031)901 1136 y Ft(0)920 1129 y Fy(\()p Fw(Y)964 1136 y Ft(0)p Fp(;)p Fk(ev)o(en)1053 1129 y Fy(\))p Fw(E)1108 1110 y Ft(+)1105 1142 y(0)1149 1129 y Fy(=)13 b(0)453 1222 y Fw(Z)484 1229 y Ft(0)504 1222 y Fw(\031)530 1229 y Ft(0)550 1222 y Fy(\()p Fw(Y)594 1229 y Ft(0)p Fp(;)p Fk(o)q(dd)673 1222 y Fy(\))41 b(=)h Fw(E)846 1203 y Ft(+)843 1234 y(0)875 1222 y Fw(\031)901 1229 y Ft(0)920 1222 y Fy(\()p Fw(Y)964 1229 y Ft(0)p Fp(;)p Fk(o)q(dd)1044 1222 y Fy(\))p Fw(E)1099 1203 y Fu(\000)1096 1234 y Ft(0)1137 1222 y Fv(\000)10 b Fw(E)1219 1203 y Fu(\000)1216 1234 y Ft(0)1248 1222 y Fw(\031)1274 1229 y Ft(0)1294 1222 y Fy(\()p Fw(Y)1338 1229 y Ft(0)p Fp(;)p Fk(o)q(dd)1417 1222 y Fy(\))p Fw(E)1472 1203 y Ft(+)1469 1234 y(0)1508 1222 y Fw(:)-12 1327 y Fy(Therefore,)17 b(applying)i(again)e(Remark)h (2.8)e(to)h(the)g(terms)1041 1314 y FB(f)1036 1327 y Fw(Z)1067 1334 y Ft(1)1087 1327 y Fv(M)p Fy(\()p Fx(q)1182 1311 y Fu(?)1182 1339 y Ft(1)1211 1327 y Fy(\))1234 1314 y FB(f)1229 1327 y Fw(Z)1260 1334 y Ft(1)1279 1296 y Fu(\003)1316 1327 y Fy(\(resp)q(ectiv)o(ely)i Fv(\000)p Fw(i)1640 1314 y FB(f)1635 1327 y Fw(Z)1666 1334 y Ft(1)1686 1327 y Fv(M)p Fy(\()p Fx(q)1781 1311 y Fu(?)1781 1339 y Ft(1)1810 1327 y Fy(\))1833 1314 y FB(f)1828 1327 y Fw(Z)1859 1334 y Ft(1)1878 1327 y Fy(\))-12 1384 y(w)o(e)c(obtain)g (that)g(the)g(tensor)g(pro)q(duct)g(\(38\))f(is)i(generated)f(b)o(y)g (the)h(elemen)o(ts)31 1489 y Fw(\031)57 1496 y Ft(0)76 1489 y Fy(\()p Fw(Y)120 1496 y Ft(0)p Fp(;)p Fk(ev)o(en)209 1489 y Fy(\))s Fv(\012)268 1442 y FB(\020)292 1489 y Fw(\031)318 1496 y Ft(1)338 1489 y Fy(\()p Fw(Y)382 1496 y Ft(1)p Fp(;)p Fk(ev)o(en)470 1489 y Fy(\))10 b(+)h Fw(iZ)591 1496 y Ft(1)610 1489 y Fw(\031)636 1496 y Ft(1)656 1489 y Fy(\()p Fw(Y)700 1496 y Ft(1)p Fp(;)p Fk(o)q(dd)779 1489 y Fy(\))797 1442 y FB(\021)834 1489 y Fy(and)i Fw(Z)951 1496 y Ft(0)971 1489 y Fw(\031)997 1496 y Ft(0)1016 1489 y Fy(\()p Fw(Y)1060 1496 y Ft(0)p Fp(;)p Fk(o)q(dd)1140 1489 y Fy(\))s Fv(\012)1199 1442 y FB(\020)1223 1489 y Fw(iZ)1270 1496 y Ft(1)1289 1489 y Fw(\031)1315 1496 y Ft(1)1335 1489 y Fy(\()p Fw(Y)1379 1496 y Ft(1)p Fp(;)p Fk(ev)o(en)1467 1489 y Fy(\))d(+)h Fw(\031)1567 1496 y Ft(1)1586 1489 y Fy(\()p Fw(Y)1630 1496 y Ft(1)p Fp(;)p Fk(o)q(dd)1710 1489 y Fy(\))1728 1442 y FB(\021)1760 1489 y Fw(;)42 b Fy(\(39\))-12 1600 y(where)15 b Fw(Y)145 1607 y Fp(k)179 1600 y Fy(=)e Fw(Y)253 1607 y Fp(k)q(;)p Fk(ev)o(en)353 1600 y Fy(+)c Fw(Y)423 1607 y Fp(k)q(;)p Fk(o)q(dd)517 1600 y Fv(2)k Fy(CAR)q(\()p Fx(q)701 1584 y Fu(?)701 1613 y Fp(k)730 1600 y Fw(;)8 b Fy(\000)779 1607 y Fp(k)800 1600 y Fy(\),)14 b Fw(k)g Fy(=)f(0)p Fw(;)8 b Fy(1.)18 b(No)o(w)c(the)h(r.h.s.)e(of)i(the)f(inclusion)j (\(37\))d(can)h(also)-12 1657 y(b)q(e)h(written)f(using)h(Prop)q (osition)g(2.4)e(and)h(Corollary)g(2.5)g(as)103 1741 y FB(e)93 1755 y Fw(Z)c Fv(M)p Fy(\()p Fx(q)230 1736 y Fu(?)259 1755 y Fy(\))294 1741 y FB(e)284 1755 y Fw(Z)318 1736 y Fu(\003)351 1755 y Fy(=)i Fv(f)p Fw(\031)r Fy(\()p Fw(Y)494 1762 y Fk(ev)o(en)555 1755 y Fy(\))c(+)i Fw(iZ)s(\031)r Fy(\()p Fw(Y)750 1762 y Fk(o)q(dd)801 1755 y Fy(\)\))h Fv(j)g Fw(Y)900 1762 y Fk(ev)o(en)972 1755 y Fy(+)f Fw(Y)1044 1762 y Fk(o)q(dd)1108 1755 y Fv(2)i Fy(CAR)q(\()p Fx(q)1292 1736 y Fu(?)1321 1755 y Fw(;)8 b Fy(\000\))p Fv(g)1411 1736 y Fu(00)135 1855 y Fy(=)212 1808 y FB(n)239 1855 y Fw(\031)265 1862 y Ft(0)285 1855 y Fy(\()p Fw(Y)329 1862 y Ft(0)p Fp(;)p Fk(ev)o(en)418 1855 y Fy(\))h Fv(\012)i Fy(1)-11 b FF(l)16 b Fy(+)10 b(1)-11 b FF(l)16 b Fv(\012)10 b Fw(\031)687 1862 y Ft(1)707 1855 y Fy(\()p Fw(Y)751 1862 y Ft(1)p Fp(;)p Fk(ev)o(en)840 1855 y Fy(\))f(+)i Fw(i)p Fy(\()p Fw(Z)978 1862 y Ft(0)1007 1855 y Fv(\012)f Fw(Z)1083 1862 y Ft(1)1103 1855 y Fy(\))1121 1808 y FB(\020)1146 1855 y Fw(\031)1172 1862 y Ft(0)1191 1855 y Fy(\()p Fw(Y)1235 1862 y Ft(0)p Fp(;)p Fk(o)q(dd)1315 1855 y Fy(\))f Fv(\012)i Fy(1)-11 b FF(l)15 b Fy(+)c Fw(Z)1504 1862 y Ft(0)1534 1855 y Fv(\012)f Fw(\031)1605 1862 y Ft(1)1625 1855 y Fy(\()p Fw(Y)1669 1862 y Ft(1)p Fp(;)p Fk(o)q(dd)1748 1855 y Fy(\))1766 1808 y FB(\021)259 1951 y Fv(j)j Fw(Y)311 1958 y Fp(k)q(;)p Fk(ev)o(en)412 1951 y Fy(+)d Fw(Y)483 1958 y Fp(k)q(;)p Fk(o)q(dd)577 1951 y Fv(2)j Fy(CAR\()p Fx(q)760 1933 y Fu(?)760 1963 y Fp(k)790 1951 y Fw(;)8 b Fy(\000)839 1958 y Fp(k)860 1951 y Fy(\))p Fw(;)g(k)k Fy(=)h(0)p Fw(;)8 b Fy(1)1050 1904 y FB(o)1077 1916 y Fu(00)135 2066 y Fy(=)212 2019 y FB(n)239 2066 y Fw(\031)265 2073 y Ft(0)285 2066 y Fy(\()p Fw(Y)329 2073 y Ft(0)p Fp(;)p Fk(ev)o(en)418 2066 y Fy(\))h Fv(\012)i Fy(1)-11 b FF(l)16 b Fy(+)10 b(1)-11 b FF(l)16 b Fv(\012)10 b Fw(\031)687 2073 y Ft(1)707 2066 y Fy(\()p Fw(Y)751 2073 y Ft(1)p Fp(;)p Fk(ev)o(en)840 2066 y Fy(\))f(+)i Fw(iZ)960 2073 y Ft(0)979 2066 y Fw(\031)1005 2073 y Ft(0)1025 2066 y Fy(\()p Fw(Y)1069 2073 y Ft(0)p Fp(;)p Fk(o)q(dd)1148 2066 y Fy(\))f Fv(\012)g Fw(Z)1252 2073 y Ft(1)1282 2066 y Fy(+)h Fw(i)p Fy(1)-11 b FF(l)15 b Fv(\012)10 b Fw(Z)1459 2073 y Ft(1)1479 2066 y Fw(\031)1505 2073 y Ft(1)1525 2066 y Fy(\()p Fw(Y)1569 2073 y Ft(1)p Fp(;)p Fk(o)q(dd)1648 2066 y Fy(\))259 2163 y Fv(j)j Fw(Y)311 2170 y Fp(k)q(;)p Fk(ev)o(en)412 2163 y Fy(+)d Fw(Y)483 2170 y Fp(k)q(;)p Fk(o)q(dd)577 2163 y Fv(2)j Fy(CAR\()p Fx(q)760 2144 y Fu(?)760 2174 y Fp(k)790 2163 y Fw(;)8 b Fy(\000)839 2170 y Fp(k)860 2163 y Fy(\))p Fw(;)g(k)k Fy(=)h(0)p Fw(;)8 b Fy(1)1050 2116 y FB(o)1077 2127 y Fu(00)1105 2163 y Fw(:)-12 2269 y Fy(Th)o(us)16 b(to)f(sho)o(w)g(the)h(inclusion)i (\(37\))c(w)o(e)i(still)h(need)f(to)f(c)o(hec)o(k)h(that)f(the)h (generators)f(\(39\))f(are)i(con)o(tained)g(in)-3 2312 y FB(e)-12 2325 y Fw(Z)11 b Fv(M)p Fy(\()p Fx(q)125 2309 y Fu(?)154 2325 y Fy(\))188 2312 y FB(e)179 2325 y Fw(Z)213 2309 y Fu(\003)233 2325 y Fy(.)20 b(Therefore)15 b(the)g(relations)170 2423 y Fw(\031)196 2430 y Ft(0)215 2423 y Fy(\()p Fw(Y)259 2430 y Ft(0)p Fp(;)p Fk(ev)o(en)348 2423 y Fy(\))10 b Fv(\012)g Fw(\031)447 2430 y Ft(1)467 2423 y Fy(\()p Fw(Y)511 2430 y Ft(1)p Fp(;)p Fk(ev)o(en)600 2423 y Fy(\))41 b(=)h(\()p Fw(\031)780 2430 y Ft(0)799 2423 y Fy(\()p Fw(Y)843 2430 y Ft(0)p Fp(;)p Fk(ev)o(en)932 2423 y Fy(\))10 b Fv(\012)g Fy(1)-10 b FF(l)5 b Fy(\))k Fv(\001)h Fy(\(1)-11 b FF(l)16 b Fv(\012)10 b Fw(\031)1214 2430 y Ft(1)1234 2423 y Fy(\()p Fw(Y)1278 2430 y Ft(1)p Fp(;)p Fk(ev)o(en)1366 2423 y Fy(\)\))i Fv(2)1467 2410 y FB(e)1457 2423 y Fw(Z)f Fv(M)p Fy(\()p Fx(q)1594 2404 y Fu(?)1623 2423 y Fy(\))1658 2410 y FB(e)1648 2423 y Fw(Z)1682 2404 y Fu(\003)128 2515 y Fw(\031)154 2522 y Ft(0)174 2515 y Fy(\()p Fw(Y)218 2522 y Ft(0)p Fp(;)p Fk(ev)o(en)307 2515 y Fy(\))f Fv(\012)g Fw(Z)411 2522 y Ft(1)431 2515 y Fw(\031)457 2522 y Ft(1)476 2515 y Fy(\()p Fw(Y)520 2522 y Ft(1)p Fp(;)p Fk(o)q(dd)600 2515 y Fy(\))41 b(=)h(\()p Fw(\031)780 2522 y Ft(0)799 2515 y Fy(\()p Fw(Y)843 2522 y Ft(0)p Fp(;)p Fk(ev)o(en)932 2515 y Fy(\))10 b Fv(\012)g Fy(1)-10 b FF(l)5 b Fy(\))k Fv(\001)h Fy(\(1)-11 b FF(l)16 b Fv(\012)10 b Fw(Z)1219 2522 y Ft(1)1239 2515 y Fw(\031)1265 2522 y Ft(1)1284 2515 y Fy(\()p Fw(Y)1328 2522 y Ft(1)p Fp(;)p Fk(o)q(dd)1408 2515 y Fy(\)\))i Fv(2)1508 2502 y FB(e)1499 2515 y Fw(Z)e Fv(M)p Fy(\()p Fx(q)1635 2497 y Fu(?)1664 2515 y Fy(\))1699 2502 y FB(e)1690 2515 y Fw(Z)1724 2497 y Fu(\003)78 2608 y Fw(Z)109 2615 y Ft(0)128 2608 y Fw(\031)154 2615 y Ft(0)174 2608 y Fy(\()p Fw(Y)218 2615 y Ft(0)p Fp(;)p Fk(o)q(dd)297 2608 y Fy(\))g Fv(\012)h Fw(Z)402 2615 y Ft(1)421 2608 y Fw(\031)447 2615 y Ft(1)467 2608 y Fy(\()p Fw(Y)511 2615 y Ft(1)p Fp(;)p Fk(ev)o(en)600 2608 y Fy(\))41 b(=)h(\()p Fw(Z)785 2615 y Ft(0)804 2608 y Fw(\031)830 2615 y Ft(0)850 2608 y Fy(\()p Fw(Y)894 2615 y Ft(0)p Fp(;)p Fk(o)q(dd)973 2608 y Fy(\))10 b Fv(\012)g Fw(Z)1077 2615 y Ft(1)1097 2608 y Fy(\))g Fv(\001)g Fy(\(1)-11 b FF(l)15 b Fv(\012)10 b Fw(\031)1276 2615 y Ft(1)1296 2608 y Fy(\()p Fw(Y)1340 2615 y Ft(1)p Fp(;)p Fk(ev)o(en)1429 2608 y Fy(\)\))i Fv(2)1529 2595 y FB(e)1520 2608 y Fw(Z)f Fv(M)p Fy(\()p Fx(q)1657 2589 y Fu(?)1686 2608 y Fy(\))1720 2595 y FB(e)1711 2608 y Fw(Z)1745 2589 y Fu(\003)138 2700 y Fw(Z)169 2707 y Ft(0)189 2700 y Fw(\031)215 2707 y Ft(0)234 2700 y Fy(\()p Fw(Y)278 2707 y Ft(0)p Fp(;)p Fk(o)q(dd)358 2700 y Fy(\))e Fv(\012)i Fw(\031)457 2707 y Ft(1)476 2700 y Fy(\()p Fw(Y)520 2707 y Ft(1)p Fp(;)p Fk(o)q(dd)600 2700 y Fy(\))41 b(=)h(\()p Fw(Z)785 2707 y Ft(0)804 2700 y Fw(\031)830 2707 y Ft(0)850 2700 y Fy(\()p Fw(Y)894 2707 y Ft(0)p Fp(;)p Fk(o)q(dd)973 2700 y Fy(\))10 b Fv(\012)g Fw(Z)1077 2707 y Ft(1)1097 2700 y Fy(\))g Fv(\001)g Fy(\(1)-11 b FF(l)15 b Fv(\012)10 b Fw(Z)1281 2707 y Ft(1)1301 2700 y Fw(\031)1327 2707 y Ft(1)1347 2700 y Fy(\()p Fw(Y)1391 2707 y Ft(1)p Fp(;)p Fk(o)q(dd)1470 2700 y Fy(\)\))i Fv(2)1570 2687 y FB(e)1561 2700 y Fw(Z)f Fv(M)p Fy(\()p Fx(q)1698 2682 y Fu(?)1727 2700 y Fy(\))1761 2687 y FB(e)1752 2700 y Fw(Z)1786 2682 y Fu(\003)-12 2798 y Fy(conclude)17 b(the)e(pro)q(of.)p 1857 2798 24 24 v 919 2967 a(22)p eop %%Page: 23 23 23 22 bop -12 50 a Fz(App)r(endix)-12 152 y Fy(W)l(e)19 b(will)h(giv)o(e)f(in)g(this)g(app)q(endix)h(the)f(pro)q(of)f(of)g (Prop)q(osition)h(2.2.)29 b(Recall)21 b(the)d(notation)g(and)h(results) g(of)-12 208 y(Section)d(2.)-12 300 y Fq(Prop)q(osition)g Fs(F)m(or)g Fw(f)347 307 y Ft(1)367 300 y Fw(;)8 b(:)g(:)g(:)d(;)j(f) 491 307 y Fp(n)526 300 y Fv(2)13 b Fx(h)j Fs(the)h(e)n(quation)167 380 y FB(\020)192 427 y Fw(a)p Fy(\()p Fw(f)256 434 y Fp(n)279 427 y Fy(\))10 b Fv(\001)g Fw(:)e(:)g(:)g Fv(\001)i Fw(a)p Fy(\()p Fw(f)480 434 y Ft(1)499 427 y Fy(\))517 380 y FB(\021)549 427 y Fy(\012)j(=)708 386 y FB(X)671 476 y Fr(\031)d Fn(2)f Fm(S)755 480 y Fl(n;)t(p)664 517 y Fk(0)g Fn(\024)f Fk(2)p Fr(p)i Fn(\024)e Fr(n)803 427 y Fy(\(sgn)f Fw(\031)r Fy(\))979 371 y Fp(p)961 386 y FB(Y)960 479 y Fp(l)p Ft(=1)1028 427 y Fv(h)p Fw(P)f(f)1103 434 y Fp(\013)1126 440 y Fr(l)1148 427 y Fw(;)15 b(P)6 b Fy(\000)p Fw(f)1261 434 y Fp(\014)1281 440 y Fr(l)1296 427 y Fv(i)i Fw(P)e Fy(\000)p Fw(f)1407 434 y Fp(j)1421 439 y Fk(1)1451 427 y Fv(^)k Fw(:)e(:)g(:)h Fv(^)h Fw(P)c Fy(\000)p Fw(f)1680 434 y Fp(j)1694 440 y Fr(k)-12 605 y Fs(holds,)16 b(wher)n(e)h(the)f(indic)n(es)f Fw(\013)502 612 y Fp(l)516 605 y Fw(;)8 b(\014)563 612 y Fp(l)574 605 y Fw(;)g(j)614 612 y Ft(1)633 605 y Fw(;)g(:)g(:)g(:)d(;)j(j)754 612 y Fp(k)791 605 y Fs(ar)n(e)16 b(given)g(in)f(the)i(de\014ntion)f (of)g Fx(S)1399 612 y Fp(n;)6 b(p)1472 605 y Fs(and)17 b(wher)n(e)f(for)h Fw(n)c Fy(=)g(2)p Fw(p)-12 662 y Fs(in)j(the)g(pr)n (e)n(c)n(e)n(ding)f(sum)h(one)g(r)n(eplac)n(es)f(the)i(we)n(dge)f(pr)n (o)n(duct)h(by)f(the)h(vacuum)g Fy(\012)p Fs(.)-12 742 y(Pr)n(o)n(of:)25 b Fy(The)19 b(pro)q(of)e(is)i(done)f(b)o(y)g (induction)i(on)d(the)h(n)o(um)o(b)q(er)h(of)e(generators)g(of)h(the)g (CAR{Algebra.)28 b(F)l(or)-12 798 y Fw(n)13 b Fy(=)g(1)g(the)h(ab)q(o)o (v)o(e)g(form)o(ula)f(is)h(immediately)i(v)o(ery\014ed)e(using)g(the)g (de\014nition)i(of)d(creation)h(and)g(annihilation)-12 855 y(op)q(erators.)34 b(Assume)20 b(that)f(it)i(holds)f(for)g Fw(n)g Fy(generators)f(and)h(w)o(e)g(pro)q(of)g(that)f(it)i(is)f(also)g (true)g(for)g Fw(n)13 b Fy(+)h(1)-12 911 y(generators.)25 b(T)l(ak)o(e)17 b Fw(f)366 918 y Fp(n)p Ft(+1)435 911 y Fw(;)8 b(f)478 918 y Fp(n)501 911 y Fw(;)g(:)g(:)g(:)t(;)g(f)624 918 y Ft(1)660 911 y Fv(2)16 b Fx(h)h Fy(and)g(from)g(the)g(preceding)i (assumption)e(as)g(w)o(ell)h(as)f(the)g(results)-12 967 y(stated)e(in)h(Section)g(2)f(w)o(e)f(ha)o(v)o(e)84 1064 y Fw(a)p Fy(\()p Fw(f)148 1071 y Fp(n)p Ft(+1)217 1064 y Fy(\))243 1017 y FB(\020)267 1064 y Fw(a)p Fy(\()p Fw(f)331 1071 y Fp(n)354 1064 y Fy(\))c Fv(\001)g Fw(:)e(:)g(:)g Fv(\001)h Fw(a)p Fy(\()p Fw(f)554 1071 y Ft(1)574 1064 y Fy(\))f(\012)633 1017 y FB(\021)285 1179 y Fy(=)362 1132 y FB(\020)386 1179 y Fw(c)p Fy(\()p Fw(P)e Fy(\000)p Fw(f)509 1186 y Fp(n)p Ft(+1)578 1179 y Fy(\))596 1160 y Fu(\003)626 1179 y Fy(+)k Fw(c)p Fy(\()p Fw(P)c(f)766 1186 y Fp(n)p Ft(+1)835 1179 y Fy(\))853 1132 y FB(\021)12 b(\020)915 1179 y Fy(\()p Fw(a)p Fy(\()p Fw(f)997 1186 y Fp(n)1020 1179 y Fy(\))e Fv(\001)g Fw(:)e(:)g(:)g Fv(\001)h Fw(a)p Fy(\()p Fw(f)1220 1186 y Ft(1)1240 1179 y Fy(\))f(\012)1299 1132 y FB(\021)285 1319 y Fy(=)427 1278 y FB(X)390 1368 y Fr(\031)i Fn(2)e Fm(S)473 1372 y Fl(n;)d(p)382 1409 y Fk(0)k Fn(\024)g Fk(2)p Fr(p)g Fn(\024)g Fr(n)522 1319 y Fy(\(sgn)e Fw(\031)r Fy(\))697 1263 y Fp(p)680 1278 y FB(Y)678 1371 y Fp(l)p Ft(=1)747 1319 y Fv(h)p Fw(P)f(f)822 1326 y Fp(\013)845 1332 y Fr(l)867 1319 y Fw(;)15 b(P)6 b Fy(\000)p Fw(f)980 1326 y Fp(\014)1000 1332 y Fr(l)1015 1319 y Fv(i)i Fw(P)e Fy(\000)p Fw(f)1126 1326 y Fp(n)p Ft(+1)1205 1319 y Fv(^)k Fw(P)c Fy(\000)p Fw(f)1330 1326 y Fp(j)1344 1331 y Fk(1)1375 1319 y Fv(^)k Fw(:)e(:)g(:)h Fv(^)h Fw(P)c Fy(\000)p Fw(f)1604 1326 y Fp(j)1618 1332 y Fr(k)1815 1319 y Fy(\(40\))362 1518 y(+)470 1478 y FB(X)433 1568 y Fr(\031)k Fn(2)e Fm(S)516 1572 y Fl(n;)d(p)425 1608 y Fk(0)k Fn(\024)g Fk(2)p Fr(p)g Fn(\024)g Fr(n)624 1466 y Fp(k)604 1478 y FB(X)602 1569 y Fp(r)q Ft(=1)690 1518 y Fy(\(sgn)e Fw(\031)r Fy(\))h(\()p Fv(\000)p Fy(1\))929 1500 y Fp(r)q Fu(\000)p Ft(1)1004 1518 y Fv(h)p Fw(P)e(f)1079 1525 y Fp(n)p Ft(+1)1155 1518 y Fw(;)15 b(P)6 b Fy(\000)p Fw(f)1268 1525 y Fp(j)1282 1529 y Fr(r)1303 1518 y Fv(i)k(\001)425 1712 y(\001)464 1656 y Fp(p)446 1672 y FB(Y)445 1764 y Fp(l)p Ft(=1)514 1712 y Fv(h)p Fw(P)c(f)589 1719 y Fp(\013)612 1725 y Fr(l)633 1712 y Fw(;)15 b(P)6 b Fy(\000)p Fw(f)746 1719 y Fp(\014)766 1725 y Fr(l)781 1712 y Fv(i)i Fw(P)e Fy(\000)p Fw(f)892 1719 y Fp(j)906 1724 y Fk(1)936 1712 y Fv(^)11 b Fw(:)d(:)g(:)g Fv(^)1110 1698 y FB(d)1080 1712 y Fw(P)e Fy(\000)p Fw(f)1165 1719 y Fp(j)1179 1723 y Fr(r)1210 1712 y Fv(^)11 b Fw(:)d(:)g(:)g Fv(^)i Fw(P)c Fy(\000)p Fw(f)1439 1719 y Fp(j)1453 1725 y Fr(k)1483 1712 y Fw(:)-12 1847 y Fy(W)l(e)17 b(will)i(determine)f(ho)o(w)f(man)o (y)g(terms)g(with)g(particle)h(n)o(um)o(b)q(er)g Fw(k)1186 1830 y Fu(0)1215 1847 y Fy(app)q(ear)f(in)h(the)g(preceding)g(sum.)26 b(F)l(or)-12 1903 y(this)14 b(let)h Fw(p)163 1887 y Fu(0)187 1903 y Fv(2)e Fg(N)f Fy(b)q(e)j(suc)o(h)f(that)f(2)p Fw(p)581 1887 y Fu(0)600 1903 y Fy(+)8 b Fw(k)668 1887 y Fu(0)693 1903 y Fy(=)13 b Fw(n)8 b Fy(+)g(1.)18 b(No)o(w)c(the)g (\014rst)f(term)h(in)g(the)g(ab)q(o)o(v)o(e)g(form)o(ula)f(con)o (tributes)-12 1960 y(b)o(y)i(means)g(of)g(expressions)h(where)f Fw(k)f Fy(=)f Fw(k)721 1943 y Fu(0)743 1960 y Fv(\000)d Fy(1)15 b(\(hence)h Fw(p)d Fy(=)g Fw(p)1077 1943 y Fu(0)1088 1960 y Fy(\))i(and)g(there)h(are)587 2017 y FB( )673 2060 y Fw(n)623 2117 y(n)10 b Fv(\000)h Fy(2)p Fw(p)754 2017 y FB(!)800 2058 y Fy(\(2)p Fw(p)p Fy(\)!)p 800 2078 94 2 v 804 2120 a Fw(p)p Fy(!)d(2)871 2107 y Fp(p)911 2089 y Fy(=)959 2017 y FB( )1051 2060 y Fw(n)995 2117 y(n)i Fv(\000)h Fy(2)p Fw(p)1124 2100 y Fu(0)1138 2017 y FB(!)1185 2058 y Fy(\(2)p Fw(p)1249 2041 y Fu(0)1260 2058 y Fy(\)!)p 1183 2078 109 2 v 1183 2120 a Fw(p)1206 2107 y Fu(0)1218 2120 y Fy(!)d(2)1262 2107 y Fp(p)1280 2097 y Fn(0)-12 2218 y Fy(suc)o(h)15 b(summands.)20 b(F)l(urther,)14 b(the)h(second)g(term)f(con)o(tributes)h(b)o(y)f(means)h(of)f (expressions)h(where)g Fw(k)f Fy(=)f Fw(k)1808 2202 y Fu(0)1829 2218 y Fy(+)c(1)-12 2275 y(\(hence)16 b Fw(p)c Fy(=)h Fw(p)238 2258 y Fu(0)260 2275 y Fv(\000)d Fy(1\))15 b(and)g(there)h(are)f(no)o(w)528 2404 y Fw(k)568 2332 y FB( )654 2375 y Fw(n)604 2432 y(n)10 b Fv(\000)h Fy(2)p Fw(p)735 2332 y FB(!)781 2373 y Fy(\(2)p Fw(p)p Fy(\)!)p 781 2393 94 2 v 785 2435 a Fw(p)p Fy(!)d(2)852 2422 y Fp(p)892 2404 y Fy(=)940 2332 y FB( )1071 2375 y Fw(n)976 2432 y(n)i Fv(\000)h Fy(2)p Fw(p)1105 2415 y Fu(0)1126 2432 y Fy(+)f(1)1197 2332 y FB(!)1244 2373 y Fy(\(2)p Fw(p)1308 2356 y Fu(0)1319 2373 y Fy(\)!)p 1243 2393 109 2 v 1243 2435 a Fw(p)1266 2422 y Fu(0)1277 2435 y Fy(!)e(2)1321 2422 y Fp(p)1339 2412 y Fn(0)-12 2530 y Fy(suc)o(h)15 b(summands.)20 b(Altogether)c(w)o(e)e(obtain)299 2584 y FB( )391 2628 y Fw(n)335 2684 y(n)c Fv(\000)g Fy(2)p Fw(p)463 2668 y Fu(0)478 2584 y FB(!)525 2626 y Fy(\(2)p Fw(p)589 2609 y Fu(0)600 2626 y Fy(\)!)p 523 2646 V 523 2687 a Fw(p)546 2674 y Fu(0)558 2687 y Fy(!)e(2)602 2674 y Fp(p)620 2665 y Fn(0)647 2656 y Fy(+)692 2584 y FB( )824 2628 y Fw(n)728 2684 y(n)i Fv(\000)h Fy(2)p Fw(p)857 2668 y Fu(0)878 2684 y Fy(+)g(1)949 2584 y FB(!)996 2626 y Fy(\(2)p Fw(p)1060 2609 y Fu(0)1071 2626 y Fy(\)!)p 995 2646 V 995 2687 a Fw(p)1018 2674 y Fu(0)1029 2687 y Fy(!)d(2)1073 2674 y Fp(p)1091 2665 y Fn(0)1121 2656 y Fy(=)1169 2584 y FB( )1261 2628 y Fw(n)j Fy(+)f(1)1205 2684 y Fw(n)g Fy(+)h(1)e Fv(\000)i Fy(2)p Fw(p)1412 2668 y Fu(0)1426 2584 y FB(!)1473 2626 y Fy(\(2)p Fw(p)1537 2609 y Fu(0)1548 2626 y Fy(\)!)p 1472 2646 V 1472 2687 a Fw(p)1495 2674 y Fu(0)1506 2687 y Fy(!)d(2)1550 2674 y Fp(p)1568 2665 y Fn(0)-12 2786 y Fy(terms)18 b(with)g(particle)i(n)o (um)o(b)q(er)e Fw(k)585 2770 y Fu(0)615 2786 y Fy(and)h(this)g (coincides)h(with)f(the)f(n)o(um)o(b)q(er)h(of)f(elemen)o(ts)h(in)g Fx(S)1689 2794 y Fp(n)p Ft(+1)p Fp(;)6 b(p)1789 2784 y Fn(0)1802 2786 y Fy(.)29 b(T)l(o)-12 2843 y(conclude)17 b(the)f(pro)q(of)f(w)o(e)g(still)i(need)f(to)f(sho)o(w)g(that)f(eac)o (h)i(term)f(in)h(the)g(sum)f(carries)h(the)f(correct)g(sign.)21 b(F)l(or)919 2967 y(23)p eop %%Page: 24 24 24 23 bop -12 50 a Fy(the)15 b(summands)g(in)i(\(40\))c(this)j(follo)o (ws)f(from)75 159 y(sgn)141 115 y FB(\020)178 139 y Fp(n)51 b(n)8 b Fu(\000)g Ft(1)42 b Fu(\001)5 b(\001)h(\001)41 b Fp(n)9 b Fu(\000)f Ft(2)p Fp(p)f Ft(+)h(2)42 b Fp(n)8 b Fu(\000)g Ft(2)p Fp(p)f Ft(+)i(1)47 b Fp(k)i Fu(\001)5 b(\001)h(\001)49 b Ft(1)168 178 y Fp(\013)191 183 y Fk(1)272 178 y Fp(\014)292 183 y Fk(1)374 178 y Fu(\001)5 b(\001)h(\001)101 b Fp(\013)539 182 y Fr(p)720 178 y Fp(\014)740 182 y Fr(p)861 178 y Fp(j)875 183 y Fk(1)934 178 y Fp(:)5 b(:)h(:)41 b(j)1030 184 y Fr(k)1053 115 y FB(\021)351 278 y Fy(=)h(sgn)494 234 y FB(\020)522 258 y Fp(n)8 b Ft(+)g(1)51 b Fp(n)g Fu(\001)6 b(\001)f(\001)49 b Ft(1)522 298 y Fp(n)8 b Ft(+)g(1)41 b Fp(\013)668 303 y Fk(1)727 298 y Fu(\001)6 b(\001)f(\001)42 b Fp(j)824 304 y Fr(k)846 234 y FB(\021)351 397 y Fy(=)g(sgn)494 353 y FB(\020)522 377 y Fp(n)8 b Ft(+)g(1)49 b Fp(n)h Fu(\001)6 b(\001)f(\001)42 b Fp(n)8 b Fu(\000)g Ft(2)p Fp(p)f Ft(+)i(3)41 b Fp(n)8 b Fu(\000)g Ft(2)p Fp(p)g Ft(+)g(2)42 b Fp(k)9 b Ft(+)g(1)48 b Fp(k)h Fu(\001)6 b(\001)f(\001)49 b Ft(1)543 417 y Fp(\013)566 422 y Fk(1)645 417 y Fp(\014)665 422 y Fk(1)724 417 y Fu(\001)6 b(\001)f(\001)102 b Fp(\013)890 421 y Fr(p)1070 417 y Fp(\014)1090 421 y Fr(p)1211 417 y Fp(n)8 b Ft(+)h(1)41 b Fp(j)1349 422 y Fk(1)1408 417 y Fp(:)6 b(:)f(:)42 b(j)1505 423 y Fr(k)1527 353 y FB(\021)1561 397 y Fw(:)-12 510 y Fy(F)l(or)15 b(the)g(remaining)h(terms)f(w)o(e)g(ha)o(v)o(e)10 616 y(\()p Fv(\000)p Fy(1\))104 597 y Fp(r)q Fu(\000)p Ft(1)180 616 y Fy(sgn)246 572 y FB(\020)283 596 y Fp(n)52 b(n)8 b Fu(\000)g Ft(1)41 b Fu(\001)6 b(\001)g(\001)47 b Fp(k)i(:)6 b(:)f(:)42 b(k)9 b Ft(+)f(2)f Fu(\000)i Fp(r)42 b(k)9 b Ft(+)g(1)e Fu(\000)h Fp(r)43 b(k)9 b Fu(\000)f Fp(r)43 b Fu(\001)5 b(\001)h(\001)49 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Fr(r)690 874 y Fp(\013)713 879 y Fk(1)815 874 y Fp(\014)835 879 y Fk(1)916 874 y Fu(\001)6 b(\001)g(\001)41 b Fp(j)1013 879 y Fk(1)1072 874 y Fu(\001)6 b(\001)g(\001)76 b Fp(j)1204 879 y Fr(r)q Fn(\000)p Fk(1)1341 874 y Fp(j)1355 879 y Fr(r)q Fk(+1)1457 874 y Fp(:)6 b(:)g(:)41 b(j)1554 880 y Fr(k)1576 810 y FB(\021)1611 854 y Fw(:)-12 967 y Fy(Therefore)15 b(w)o(e)g(ha)o(v)o(e)g(sho)o(wn)g(that)113 1055 y FB(\020)137 1102 y Fw(a)p Fy(\()p Fw(f)201 1109 y Fp(n)p Ft(+1)270 1102 y Fy(\))10 b Fv(\001)f Fw(:)f(:)g(:)h Fv(\001)g Fw(a)p Fy(\()p Fw(f)470 1109 y Ft(1)490 1102 y Fy(\))508 1055 y FB(\021)540 1102 y Fy(\012)j(=)731 1062 y FB(X)670 1151 y Fr(\031)e Fn(2)f Fm(S)754 1160 y Fl(n)p Fb(+1)p Fl(;)c(p)837 1154 y Fe(0)654 1194 y Fk(0)k Fn(\024)g Fk(2)p Fr(p)742 1183 y Fe(0)762 1194 y Fn(\024)g Fr(n)e Fk(+)h(1)857 1102 y Fy(\(sgn)g Fw(\031)r Fy(\))1028 1046 y Fp(p)1046 1035 y Fn(0)1016 1062 y FB(Y)1014 1154 y Fp(l)p Ft(=1)1083 1102 y Fv(h)p Fw(P)e(f)1158 1109 y Fp(\013)1181 1115 y Fr(l)1203 1102 y Fw(;)14 b(P)6 b Fy(\000)p Fw(f)1315 1109 y Fp(\014)1335 1115 y Fr(l)1350 1102 y Fv(i)i Fw(P)e Fy(\000)p Fw(f)1461 1109 y Fp(j)1475 1114 y Fk(1)1506 1102 y Fv(^)k Fw(:)e(:)g(:)g Fv(^)i Fw(P)c Fy(\000)p Fw(f)1734 1109 y Fp(j)1748 1115 y Fr(k)-12 1288 y Fy(whic)o(h)16 b(concludes)h(the)e(pro)q(of.)p 1857 1288 24 24 v -12 1443 a Fa(Ac)o(kno)o(wledgemen)o(ts)43 b Fy(It)14 b(is)g(a)f(pleasure)h(to)f(thank)g(Dr.)g(H.)g(Neidhardt)h (for)f(discussions)i(on)f(the)f(sub)s(ject,)-12 1500 y(in)j(particular)f(on)g(Prop)q(osition)h(4.7)e(and)h(Theorem)g(4.18.)j (One)e(of)f(us)g(\(F.Ll.\))k(w)o(ould)d(also)f(lik)o(e)h(to)e(thank)h (a)-12 1556 y(useful)h(con)o(v)o(ersation)f(with)h(Dr.)e(C.)g (Binnenhei)k(in)e(G\177)-23 b(ottingen.)-12 1700 y Fz(References)11 1801 y Fy([1])21 b(N.I.)13 b(Ac)o(hieser)g(and)g(I.M.)f(Glasmann,)h (Theorie)g(der)g(linearen)h(Op)q(eratoren)f(im)g(Hilb)q(ert-Raum,)h(V)l (erlag)81 1857 y(Harri)i(Deutsc)o(h,)e(Th)o(un,)h(1981.)11 1951 y([2])21 b(H.)e(Araki,)g(A)f(lattice)i(of)e(v)o(on)g(Neumann)h (algebras)g(asso)q(ciated)f(with)h(the)g(quan)o(tum)f(theory)g(of)h (free)81 2008 y(b)q(ose)d(\014eld,)g Fs(J.)g(Math.)h(Phys.)d Fa(4)i Fy(\(1963\),)d(1343{1362.)11 2102 y([3])p 81 2102 137 2 v 165 w(,)19 b(On)h(quasifree)f(states)f(of)g(CAR)h(and)f (Bogoliub)q(o)o(v)i(automorphisms,)f Fs(Publ.)g(RIMS,)f(Kyoto)81 2158 y(Univ.)d Fa(6)g Fy(\(1970/71\),)d(385{442.)11 2252 y([4])p 81 2252 V 165 w(,)h(Bogoljub)q(o)o(v)h(automorphisms)e(and)h(F) l(o)q(c)o(k)g(represen)o(tations)g(of)g(canonical)h(an)o(ticomm)o (utation)81 2308 y(relations,)21 b(in)g("Op)q(erator)e(Algebras)h(and)g (Mathematical)f(Ph)o(ysics,)i(\(Pro)q(ceedings)g(of)e(the)h(summer)81 2365 y(conference)d(held)g(at)e(the)h(Univ)o(ersit)o(y)g(of)g(Io)o(w)o (a,)e(1985\)",)g(P)l(.E.T.)h(Jorgensen)h(and)g(P)l(.S.)f(Muhly)i (\(eds.\),)81 2421 y(American)f(Mathematical)f(So)q(ciet)o(y)l(,)h(Pro) o(vidence,)g(1987.)11 2515 y([5])21 b(J.)d(Avron,)f(R.)h(Seiler,)h(and) e(B.)h(Simon,)g(Charge)f(de\014ciency)l(,)j(c)o(harge)d(transp)q(ort)f (and)i(comparison)f(of)81 2571 y(dimensions,)g Fs(Commun.)f(Math.)h (Phys.)d Fa(159)i Fy(\(1994\),)d(399{422.)11 2665 y([6])p 81 2665 V 165 w(,)i(The)h(index)g(of)f(a)g(pair)g(of)g(pro)s(jections,) g Fs(J.)h(F)m(unct.)f(A)o(nal.)f Fa(120)i Fy(\(1994\),)d(220{237.)11 2759 y([7])21 b(H.)14 b(Baumg\177)-23 b(artel,)13 b(Op)q (eratoralgebraic)h(Metho)q(ds)g(in)h(Quan)o(tum)e(Field)i(Theory)l(.)f (A)f(Series)i(of)e(Lectures,)81 2816 y(Ak)m(ademie)k(V)l(erlag,)e (Berlin,)i(1995.)919 2967 y(24)p eop %%Page: 25 25 25 24 bop 11 50 a Fy([8])21 b(H.)f(Baumg\177)-23 b(artel,)20 b(M.)e(Jurk)o(e,)j(and)f(F.)f(Lled\023)-23 b(o,)21 b(On)f(free)g(nets)f (o)o(v)o(er)g(Mink)o(o)o(wski)h(space,)g Fs(R)n(ep.)g(Math.)81 107 y(Phys.)15 b Fa(35)h Fy(\(1995\),)d(101{127.)11 199 y([9])21 b(H.)11 b(Baumg\177)-23 b(artel)10 b(and)h(M.)f(W)l(ollen)o(b) q(erg,)i(Causal)f(Nets)f(of)h(Op)q(erator)f(Algebras.)h(Mathematical)f (Asp)q(ects)81 256 y(of)15 b(Algebraic)i(Quan)o(tum)e(Field)h(Theory,)f (Ak)m(ademie)i(V)l(erlag,)e(Berlin,)h(1992.)-12 348 y([10])21 b(J.J.)16 b(Bisognano)f(and)h(E.H.)f(Wic)o(hmann,)g(On)h(the)g(dualit)o (y)g(condition)h(for)e(quan)o(tum)g(\014elds,)h Fs(J.)g(Math.)81 405 y(Phys.)f Fa(17)h Fy(\(1976\),)d(303{321.)-12 497 y([11])21 b(S.)15 b(Borac,)g(On)g(the)g(algebra)g(generated)h(b)o(y)f (t)o(w)o(o)e(pro)s(jections,)i Fs(J.)h(Math.)g(Phys.)f Fa(36)g Fy(\(1995\),)e(863{874.)-12 590 y([12])21 b(O.)c(Bratteli)f (and)h(D.W.)e(Robinson,)i(Op)q(erator)g(Algebras)f(and)h(Quan)o(tum)f (Statistical)h(Mec)o(hanics)g(1,)81 647 y(Springer)f(V)l(erlag,)g (Berlin,)g(1987.)-12 739 y([13])21 b(C.)15 b(Da)o(vis,)f(Separation)i (of)f(t)o(w)o(o)f(linear)i(subspaces,)f Fs(A)n(cta)h(Sci.)g(Math.)g (Sze)n(ge)n(d)j Fa(19)c Fy(\(1958\),)e(172{187.)-12 832 y([14])21 b(G.F.)14 b(Dell'An)o(tonio,)h(Structure)f(of)h(the)f (algebras)h(of)f(some)g(free)h(systems,)f Fs(Commun.)h(Math.)h(Phys.)f Fa(9)81 888 y Fy(\(1968\),)e(81{117.)-12 981 y([15])21 b(J.)d(Dixmier,)h(P)o(osition)e(relativ)o(e)i(de)f(deux)g(v)m(ari)o (\023)-21 b(et)o(\023)g(es)16 b(lin)o(\023)-21 b(eaires)19 b(ferm)o(\023)-21 b(ees)17 b(dans)g(un)h(espace)h(de)f(Hilb)q(ert,)81 1037 y Fs(R)n(ev.)e(Sci.)f Fa(86)g Fy(\(1948\),)e(387{399.)-12 1130 y([16])21 b(S.)11 b(Doplic)o(her,)h(R.)e(Haag,)h(and)f(J.E.)g(Rob) q(erts,)i(Lo)q(cal)f(observ)m(ables)h(and)e(particle)i(statistics)e(I,) h Fs(Commun.)81 1187 y(Math.)17 b(Phys.)e Fa(23)g Fy(\(1971\),)e (199{230.)-12 1279 y([17])p 81 1279 137 2 v 165 w(,)f(Lo)q(cal)h (observ)m(ables)g(and)f(particle)h(statistics)f(I)q(I,)g Fs(Commun.)h(Math.)h(Phys.)d Fa(35)h Fy(\(1974\),)e(49{85.)-12 1372 y([18])21 b(J.P)l(.)d(Ec)o(kmann)h(and)f(K.)h(Osterw)o(alder,)f (An)h(application)h(of)e(T)l(omita's)f(theory)h(of)g(mo)q(dular)h(Hilb) q(ert)81 1428 y(algebras:)h(dualit)o(y)c(for)f(free)g(b)q(ose)h (\014elds,)g Fs(J.)g(F)m(unct.)f(A)o(nal.)f Fa(13)i Fy(\(1973\),)d (1{12.)-12 1521 y([19])21 b(D.E.)g(Ev)m(ans)i(and)f(Y.)g(Ka)o(w)o (ahigashi,)h(Quan)o(tum)f(Symmetries)h(and)f(Op)q(erator)g(Algebras,)i (Oxford)81 1577 y(Science)17 b(Publications,)g(Claredon)f(Press,)e (Oxford,)h(1998.)-12 1670 y([20])21 b(J.J.)14 b(F)l(oit,)f(Abstract)f (t)o(wisted)i(dualit)o(y)g(for)e(free)i(F)l(ermi)g(\014elds,)g Fs(Publ.)g(RIMS,)g(Kyoto)h(Univ.)d Fa(19)i Fy(\(1983\),)81 1726 y(729{741.)-12 1819 y([21])21 b(R.)16 b(Haag,)e(Lo)q(cal)i(Quan)o (tum)f(Ph)o(ysics,)g(Springer)i(V)l(erlag,)e(Berlin,)h(1992.)-12 1912 y([22])21 b(P)l(.R.)15 b(Halmos,)g(Tw)o(o)f(subspaces,)i Fs(T)m(r)n(ansactions)e(A)o(m.)i(Math.)g(So)n(c.)f Fa(144)g Fy(\(1969\),)f(381{389.)-12 2004 y([23])21 b(P)l(.D.)e(Hislop,)i(A)f (simple)h(pro)q(of)e(of)g(dualit)o(y)h(for)f(lo)q(cal)h(algebras)g(in)g (free)f(quan)o(tum)g(\014eld)i(theory,)f Fs(J.)81 2061 y(Math.)d(Phys.)e Fa(27)g Fy(\(1986\),)e(2542{2550.)-12 2153 y([24])21 b(M.)d(Jurk)o(e,)h(Ergebnisse)g(zu)g(massiv)o(en,)g (freien)g(Netzen)h(\177)-24 b(ub)q(er)19 b(dem)g(Mink)o(o)o(wskiraum,)f (Ph.D.)f(thesis,)81 2210 y(Univ)o(ersit\177)-23 b(at)16 b(P)o(otsdam,)e(1997.)-12 2303 y([25])21 b(R.V.)g(Kadison)g(and)g(J.R.) g(Ringrose,)h(F)l(undamen)o(tals)f(of)f(the)h(Theory)f(of)h(Op)q (erator)f(Algebras)h Fw(I)t(I)t Fy(,)81 2359 y(Academic)c(Press,)d (Orlando,)i(1986.)-12 2452 y([26])21 b(T.)15 b(Kato,)f(P)o(erturbation) h(Theory)g(for)g(Linear)h(Op)q(erators,)e(Springer)i(V)l(erlag,)g (Berlin,)g(1995.)-12 2544 y([27])21 b(P)l(.)16 b(Leyland,)h(J.E.)e(Rob) q(erts,)h(and)g(D.)g(T)l(estard,)f(Dualit)o(y)h(for)f(quan)o(tum)h (free)g(\014elds,)h(preprin)o(t,)f(CNRS)81 2601 y(Marseille,)h(1978.) -12 2693 y([28])k(F.)d(Lled\023)-23 b(o,)21 b(Algebraic)f(prop)q (erties)g(of)e(massless)h(free)g(nets,)g(Ph.D.)f(thesis,)i(Univ)o (ersit)o(y)g(of)e(P)o(otsdam,)81 2750 y(1998.)-12 2843 y([29])j(K.)16 b(Osterw)o(alder,)e(Dualit)o(y)i(for)f(free)g(b)q(ose)g (\014elds,)h Fs(Commun.)h(Math.)f(Phys.)f Fa(29)g Fy(\(1973\),)e(1{14.) 919 2967 y(25)p eop %%Page: 26 26 26 25 bop -12 50 a Fy([30])21 b(M.A.)16 b(Rie\013el)j(and)e(A.)g(v)m (an)g(Daele,)h(A)f(b)q(ounded)h(op)q(erator)e(approac)o(h)h(to)f(T)l (omita{T)l(ak)o(esaki)g(theory,)81 107 y Fs(Paci\014c)g(J.)g(Math.)f Fa(69)h Fy(\(1977\),)d(187{221.)-12 200 y([31])21 b(S.J.)15 b(Summers,)g(Normal)g(pro)q(duct)g(states)f(for)g(fermions)h(and)g(t)o (wisted)g(dualit)o(y)h(for)e(CCR-)h(and)g(CAR-)81 257 y(t)o(yp)q(e)f(algebras)f(with)h(applications)h(to)d(Y)l(uk)m(a)o(w)o (a)916 264 y Ft(2)949 257 y Fy(quan)o(tum)h(\014eld)i(mo)q(del,)f Fs(Commun.)h(Math.)f(Phys.)f Fa(86)81 313 y Fy(\(1982\),)g(111{141.)-12 407 y([32])21 b(M.)15 b(T)l(ak)o(esaki,)g(Theory)g(of)f(Op)q(erator)h (Algebras)h(I,)f(Springer)h(V)l(erlag,)f(New)g(Y)l(ork,)g(1979.)919 2967 y(26)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9907080444296--