Operator Algebras in Dynamical Systems (Encyclopedia of Mathematics and its Applications 41)

{l-a^-V)}-1'2"^ Z ( - ^ " ^ ^ ^ ^ ^ ^ nl

The right side of this equality is uniformly convergent on {zeC||z| < /?} and is Mfi-hj2)}" analytic on {zeC||z|
and

II vjf.vj* - fj|| < || fjbjfdl

- / i + V ) " 1/2(1 - / i + hjrll2fibj*fj-

fjII

- A + v r 1 - n/iV/.n + Wfjbjftffj-fjbjf^jW + Wfjbjteu-ej^jW + Wej^j-fjW < | | ( 1 - A + / , / ) " ' - 1 | | +9n!(21)V <||(l-/1+/i/)-1-l||+n!(21)"+1e'j* ^ /,- and ^-/i u/ is a projection, ^ / j t)j* = /,-. Put Wj- = ^-/j (j = 1,2,..., n); then w^eA(<5), and w^Wy = / x and Wjwf — fj. Put fu = w;w;*; then { / y | i j = 1,2,...,n} is a matrix unit such that f{jSA{d). Moreover ll«y-/yll = ll«a«ji*-w,w/|| ^ ||e tl e n * - wten* \\ + ||w,(e n *-w/)|| en fjbjf, - vjf, || < 3n!(21)"e' + || ht - f, \ 3n!(21)"£' + || hj - A K 6n!(21)"e' + || h,2 - fx || for 1,7 = 1,2, ...,n.

D

Proof of Theorem 4.5.1 Let {£„} be an increasing sequence of finite type I subfactors in A such that leB n , B , c B , t l and (J^°=1Bn is dense in A. Let {cy | i, j = 1,2,..., nx} be a matrix unit of Bt; then for e > 0 there is a matrix unit {fij\ij = 1,2,...,^} in A{8) such that ||e V } - f t j | | <s/n12 (i,j = 1,2,...,n t ). Let A1 be the type IBl subfactor of A generated by {fij\i,j=\,2,...,n1}.

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Operator algebras in dynamical systems

Then clearly sup

inf || x —3; || < e .

Since A1®A'1=A, where A\ is the commutant of A x in A and A\ is *-isomorphic to B'v there is a finite type Im-subfactor N in A\ such that sup

inf

xeB2

yeAi®N

|| x — j || < e/4

Since A(S) = A1 (g)(A(5)nA\\ A\nA(S) is dense in A\\ hence by the above lemma, there is a finite type Im-subfactor Nx in A\r\A(5) such that sup

inf ||x —3;|| <e/4(n 1 ) 2

xeN

yeNi

Then sup

inf

X6B2

ye>li ® Afi

|| x - y || < e/2.

Put i4x ® Ni = A 2 . Continuing this process one can construct an increasing sequence {An} of finite type-I subfactors in A such that leAn, AnaAn+l (n = 1,2,...). Moreover, sup

inf || x — y\\ <e/n and AnczA(3).

xeBn

yeAn

Hence (J^° =1 ^ n is dense in A.

•

By a minor change of the above discussions, one can easily prove the following corollary. 4.5.3 Corollary Let S be a closed *-derivation in a UHF algebra A; then there is an increasing sequence {An} of finite type-I subfactors in Q){&) such that leAn, AnaAn+l and [J™=1An is dense in A. Now we shall define a normal *-derivation in UHF algebras more restrictively than in general cases. 4.5.4 Definition Let S be a ^-derivation in a UHF algebra A. d is said to be normal if there is an increasing sequence {An} offinite type I subfactors in A such that leAn, AnczAn+1 and 2>{8) = Un°°=i^From 4.5.1 and 4.5.3, one can see that a normal *-derivation in a UHF algebra will play a key role in the study of unbounded derivations in UHF algebras. Most unbounded derivations in quantum lattice systems and Fermion field theory have normal *-derivations as their cores.

4.5 UHF algebras and normal *-derivations

159

Let S be a normal *-derivation in a UHF algebra A and let Q)(b) = \J™= x An. Let {etj\ij= 1,2,..., n x } be a matrix unit of Ax. Set

then one can easily see that d(a) = [ift l9 a] for aeAx. Since <M£ii)£i; I =

>

etidie:]

)=

>

an

^ii^(^ii)

d

«1

^i^ii

)=

Z ^i)^ii +

hence i= 1

i= 1

Therefore fex is a self-adjoint element in A. Similarly we have a sequence (hn) of self-adjoint elements in A such that (5(a) = '\[hn,a] (aeAn) (n= 1,2,...,). Moreover if S is a generator and An a A(S\ then hneA(S). 4.5.5 Proposition Let S be a normal *-derivation in a UHF algebra A such that @(d)= {J™=lAn. Then for £ > 0 , there is a normal ^-derivation dE in A such that @(5e)=\J?=1An9 de(^(Se))cz^(dE) ( n = l , 2 , . . . , ) and S-dE is a bounded ^-derivation in A with \\S — Se\\ < s.

Proof Let x be the unique tracial state on A and let Pn be the canonical conditional expectation of A onto An defined by i(xy) = T(Pn(x)y) for xeA and yeAn. Let S{a) = i[hn9a] (aeAn). Take Pni with nx ^ 1 such that \\hx -P n i (fei)|| < e/2 2 and PM2 with n2 ^ 2 such that || (/i2 - ftj - Pn2(h2 - hx) \\ < s/23. Continuing this process, take Pn. with rij>j such that

Put /j. = (hj -hj.J-

Pn.(hj - hj.i); then 2J°= i II0II < e/2, where h0 = 0, and

so XJ1 ih = d is a n element of A. Moreover for aeAjo9

=i|

[ ( ^ - /!,._,), a]-if,

LPnj(hj - V i). «]•

Fori>;0, [P n .(^ - ^._ J,a] = (Pn.(hj - hj.^a - a(Pn.(hj - hj.,))

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Operator algebras in dynamical systems

Hence

. ]

{.

Now put <5e = <5 — <5id; then and

®0e) = 0 4 .

Se(a) = i

»=1

I Pnjihj-hj.^a

(aeAj0)

Lj=l

J

a n d s o <5£(^(<5£)) c ® ( 5 e ) a n d || (5 - Se \\ = \\ Sid \ \ < s .

•

Remark This proposition implies that a normal *-derivation can be perturbed to a normal *-derivation with a finite range interaction by a bounded ""-derivation. Since bounded perturbations do not change many physical phenomena (for example, phase transition [(cf. 4.4,4.7)]), one may reduce the study of normal *-derivations to the one of normal *-derivations with finite range interaction in many cases. 4.5.6 Proposition Let S be a normal ^-derivation in a UHF algebra A with d(Q)(d)) a Q)(d). Then there is an increasing sequence An of finite type I subfactors in @(S) and two normal *-derivations c^ and S2 such that ^(<5i) = 2{52\ SMiJ^Am ( n = l , 2 , . . . , ) , S2(A2n+l) cz A2n+1 ( n = l , 2 , . . . , ) and S = SX +52. Proof Let ®(d)={J™=1Bn;

then ^ B J c B ^ , 8(Bni)cBH2,...9

and BB1 c

Set Bn. = A-, then (5(^) c ^ j + x (7 = 1 , 2 , . . , ) . Let 5(a) = i[fcB, a] (aeAJ, and define

[

and

[

v^

~l

^1+ L ( /l 2n+l-^2n + l(^2n+l))^ n=l J

00

~~|

/

«=1

J

\

00

-i

For n= 1

since hnsAn+1.

For

n+l))^ N J

aeA2no+1, r

no

"I

L« = i

J

n= 1

4.5 UHF algebras and normal *-derivations

161

Moreover if aeA2no, then

2no+l(^2n o +l)-J

If e 4 2 n o + 1 , t h e n 0e4 2 l l o + 2; hence; (<5i + <52)(a) = (5(4

•

4.5.7 Definition Let S be a normal ^-derivation in a UHF algebra A. 3 is said to be commutative if one can choose a mutually commuting family {hn} of self-adjoint elements in A and an increasing sequence {An} of finite type I subfactors in A such that \J™=1An = @(8) and 5(a) = i[hn,a] (aeAn) ("=1,2,...,). Remark The notion of commutative normal *-derivations is a generalization of classical lattice systems. In 4.5.6, (51(^2n) ^ A2n (resp. S2(A2n+1) c A2n + 1); therefore, one can choose a sequence {h2n} (resp. {h2n+ x}) of self-adjoint elements in A such that h2neA2n (resp. h2n+leA2n+1) and d1(a) = i[h2n,d] (aeA2n) (resp. <52(a) = i|7i 2 n + 1 ,a] (aeA2n+1)). Since h2in+1)-h2neA2(n+1)nA'2n (resp. h2{n+1) + 1-h2n+1e ^2(n+i)+i^^2n + i)' ih2n} (resp. {h2n+1}) is a mutually commuting family. Hence Sx (resp. 32) is commutative. Therefore 4.5.5 and 4.5.6 imply that any normal *-derivation can be written as a sum of two commutative normal *-derivations after bounded perturbation. On the other hand, bounded perturbations will not change many important physical phenomena. For example, a phase transition will not be changed by bounded perturbation (cf. 4.4, 4.7). Moreover one can analyse the commutative normal derivations than more easily the general normal derivations (cf. 4.6). Therefore, 4.5.6 may be useful for the study of normal derivations. A more powerful tool would be supplied if a normal ""-derivation becomes a commutative normal *-derivation after bounded perturbation. Therefore an answer to the following problems would be interesting. 4.5.8 Problem Let S be a normal ^-derivation in a UHF algebra A. Then can one choose normal *-derivations S1 and S2 in A such that Q){b) = @(&i) = @(82), and 6 = (5X + <52, with S1 commutative and 52 bounded! In the following, we shall show that all C*-dynamics appearing in quantum lattice systems and Fermion field theory are approximately inner. Let S be

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Operator algebras in dynamical systems

a normal *-derivation in a UHF algebra A with 3(3) = \J™=lAn. Let {hn} be a sequence of self-adjoint elements in A such that 3(a) = i[/in, a] (aeAn). Then by 3.2.22, 3 is well-behaved and so || (1 ± S)(a) \\>\\a \\ (ae3(3)) (3.2.19). Therefore, 3 is a pre-generator if and only if (1 + 3)3(3) are dense in A. In quantum lattice systems, it is not so easy to check the density of (1 + 3)3(3) directly; instead, the analytic method is often used. Suppose that 3(3(3)) cz 3(3) (finite range interaction); then one can consider iterations 3n (n = 1,2,3,...,) on 3(3). An element a in 3(3) is said to be analytic with respect to 3 if there exists a positive number r such that Z^°=o( II ^"(a) ||/it!)r" < + oo (3°(a) = a). Let A(3) be the set of all analytic elements in 3(3). If A(3) is dense in A, then by 3.4.5, 3 is a pre-generator. If a quantum lattice system has a translation-invariant, finite range interaction, then A(3) = 3(3) and so 3 is a pre-generator (cf. [159]). In particular, the Examples (4) and (5) in §3.1 are pre-generators. For more general normal *-derivations, we shall use bounded perturbations (4.5.5). Since the property of 'pre-generator' is invariant under bounded perturbations, it is sufficient to study the generation problem for normal *-derivations under the assumption of 3(3(3)) c 3(3). Next suppose that a normal *-derivation 3 in A is a pre-generator; then

11(1 ±3ihny1(l±3)a-(l±3)-1(l±3)a\\^

||(1 ±3ihny1{(l±3)a-(l±3ihn)}a\\ (aeAn). ^\\(3-3ihn)(a)\\=0

Hence (1 ±3ihn)~ 1-+(1±3)~1 (strongly) so that exp(t3) = stronglimexp(tdihn) namely {exp(£<5)|felR} is approximately inner. Therefore it is approximately inner whenever a quantum lattice system defines a C*-dynamics. Next, we shall consider quasi-free derivations in the Example (6) in §3.1. Let if be a self-adjoint operator in a Hilbert space Jf and let H = Hx + T be the Weyl decomposition, where Hj is a diagonalizable self-adjoint operator and T is of Hilbert-Schmidt class; then

Let and dim(£;)= 1 (ij = 1,2,...,). Let Pn = Zni=1Ei and Jtro = \J^1Pnje. 3 be the restriction of 3iH to ja/ 0 (Jf 0 ); then 3 is a normal *-derivation. Since H\3tf0 = H, d = dlH9 and moreover (1 ± 3)3(3) => a((l ± \H)(3(H))) = a(Jtf); hence (1 ± 3)3(3) = j / p f ) , and so 3iH is a pre-generator and {Qxp(t3iH)\teR} is approximately inner. More generally, if S is a symmetric operator, we can easily see that (5iS has a normal *-derivation 3 as a core (i.e. 3 = 3iS\ by using the polar decomposition of S. If S has no self-adjoint extension, then 3iS does not necessarily have a generator extension. We have seen that all C*-dynamical generators appearing in quantum lattice systems and Fermion field theory are approximately inner. On the

4.5 UHF algebras and normal *-derivations

163

other hand, if a C*-dynamics {A, a} (A, UHF algebra) is approximately inner, then it has many physical properties, so that a physical theory can be developed for approximately inner dynamics. Therefore the following conjecture is very important. 4.5.9 The Powers-Sakai conjecture Any C*-dynamics {A, a} with a UHF algebra A is approximately inner. Let (xt = Qxp(tS); then the following problem is important. 4.5.10 The core problem Does every generator 3 have a normal *-derivation (5X as a core (i.e. 8X = 3)1 An affirmative solution to the core problem would imply an affirmative solution to the conjecture. Therefore the affirmative solution to the core problem would be more desirable than the affirmative solution to the conjecture from the point of view of quantum lattice systems. In the following, we shall discuss both the conjecture and the core problem. Let a, = exp(f^). From 4.5.1, there is an increasing sequence {An} of finite type I subfactors in A such that AncAH + l9 \J?=1An a A(S) and \J™=1An is dense in A. Let {hn} be a sequence of self-adjoint elements in A such that d(a) = \[hn,a] (aeAn)(n= 1,2,3,...,). If (1 -8)[J™=lAn is dense in A, then 8 is the closure of restriction of 3 to \J™=1An and so has a normal *-derivation as a core; hence exp(td) = strong lim exp(t8ihn) - namely a is approximately inner. However, from the considerations on quasi-free derivations, one can easily see that the density of (1 — 8){J™=1An cannot be expected under a general selection of {J™=1An. An important, difficult problem is whether or not one can choose {An} such that (1 — 8)[J™=1An is dense in A. On the other hand, if the conjecture is true, then a general {An} may have all the necessary information to prove it in a sense. In fact, if (1 — 8iln)~ x ->(1 — 8)~1 (strongly) for some (/„), then by the density of \J™=1An, there is a sequence (sn) of such that ||/w — sn\\ < 1/n. Then, self-adjoint elements in [j^1An

1

- ( 1 - ^ i i j " 1 } ^ ) ! ! + II{(1 - ^ t i j " 1 - ( 1 - ^

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Operator algebras in dynamical systems

Hence there is a sequence {sn} in \J™=1An such that (l-S.J-^il-Sy1

(strongly).

4.5.11 Proposition (l-diln)~1-^(l-S)~1 (strongly) if and only if for each ae@(8\ there is a sequence {an} in A such that an^a and Siln(an)-+d(a). Proof Suppose that (l-5ilny1^(l-Sy1; then Siln(l -Silny1=(l-Silny1 1 l - x S ( l -Sy (strongly). F o r ae2(8\ t a k e beA such t h a t a = (l-d)~1b (note {\-5y1A = Qj(d)\ and put an = (\-5Un)~1b\ then an^a and )' Conversely, for ae@{8),

- <W~ \1 - Siln)an - a|| + ||(1 - SuX1 {8iln(aH)- S(a) + a-an} \

Since (1 - 8)2(8) = A, (1 - 8Un)'* ^ ( 1 - 8)'' (strongly). Now take xnG[]^xAr and \\8(a)-

such that || xn-an

D

\\ 0

Siln(xn) \\ ^ \\ 8(a) - Siln(an) || + || Siln(xn) - Siln(an) || -> 0. Therefore

we have: 4.5.12 Proposition {exp(td)\teU} is approximately inner if and only if for a general increasing sequence {An} of finite type I subfactors in A with U™=1An = A, there is a sequence {sn} of self-adjoint elements in \J™=1An such that for each ae@(8)9 there exists a sequence (an) in \J^=1An with an^a and

However this proposition is not so useful, because it does not suggest how to construct {sn} and {an}. In the following, we shall formulate a new problem to attack the conjecture. Let $) = \J™=1An9 where {An} is the one given in 4.5.1. Since 2> a A{S), n 6 (Q>) a A(S) (n = 1,2,3,...,). Let B be the linear subspace of A spanned by

4.5.13 Proposition (1 — S)B is dense in A. Proof Suppose that (1 — S)B is not dense in A; then there is an element / in the dual A* of A such that / ( ( I - S)B) = 0. Therefore f(x) = f(S(x))(xeB) and so f{a) = f{5m{a)){ae\J™=1An\

r > 0 such that

m= 1,2,3,...,). Since aeA(S), there is an

4.5 UHF algebras and normal *-derivations

/(«,(«)) = 1 f ^ ~ m! m=o

?" = exp(t)/(fl)

165

(111 ^ r).

t->/(a,(a)) and t-»exp(t)/(a) are real analytic on R; hence /(a t (a)) = exp(t)/(a)

(teR;ae

(J 4 , ).

I/(«,(«))I < II / II II a ||; hence /(a) = 0 and so / = 0.

•

4.5.14 Proposition Let {kn} be a sequence of self-adjoint elements in A and suppose that for each aeB, there is a sequence {an} of elements in A such that an-+a and Sikn(an)^S(a); then exp(t8) = strong limexp(t(5ifcn). Proof II (1 " «*„)- X(l - S)(a) - (1 - Sy \l - S){a) || - || (1 - Sikny \1 - S)(a) - a \\ = II (1 ~ ^ J " '{(1 - *iJM

+ (1 " S)(a) - (1 - Sikn)(an)} - a \\

Hence by 4.5.13 and 4.1.2, exp(td) = strong lim exp(fr5ifcJ.

D

4.5.15 Corollary If there is a sequence {kn} of self-adjoint elements in A such that for ae[j^lAn,dm{a) = \imn^J.xknm{a){m=\^\...X then exp{tS) = strong lim Qxp(t3ikn). Proof Siknm(a)^3m{a) and dikn(Siknm(a))->S(dm(a))(m = 0,1,2,...,). Hence for aeB there is a sequence of elements {an} in A such that an^a and Sikn(an) -+ S(a). By 4.5.14, strong lim exp(r<5ikn) = exp(^). D An important fact is that all generators considered in Examples (4)-(6) in §3.1 satisfy the assumption of this corollary. For completeness, we shall show that fact below for quasi-free derivations in which case the assertion may not be trivial. Let Jf be a Hilbert space and H be a self-adjoint operator in Jf7. Let A(H) be the set of all analytic vectors in J^ with respect to if, and let {Vn} be an increasing sequence of finite-dimensional subspaces of A(H) such that (J * =1 Vn is dense in 3tf. Let En = the linear span of |J" m=0 H m F n where H°Vn = Vn\ then En is finite-dimensional and so s/(En) is a full matrix algebra; hence there is an element kn in ^/(Jf) such that SiH(a) = '\[kn,a~\(aestf(En))\

166

Operator algebras in dynamical systems

then <SiH» = 3iknm(a)(ae^(Vn);

m = 1,2,..., n; n = 1,2,3,...,).

Therefore Siknm(a)^Sm{a)(n-^ oo;m= 1,2,...,) for aG(Jn°° If S([j^=1An) a [j^=1An; then one can easily see that the assumptions of 4.5.15 are satisfied. Therefore if there is an increasing sequence {An} such that [J:=1An cz ,4(5), d({J?=1An) cz U ^ * and U n °° = 1 ^ is dense in A, then all the assumptions of 4.5.15 are satisfied. By 4.5.5, one can change S to Sx by a bounded perturbation such that 81([j^=1An) a \J™=1An. However bounded perturbations do not generally preserve the analyticity - namely, Therefore the following open problem is generally \J^=1Anc^A(81). interesting. 4.5.16 Problem Let S be a generator in a UHF algebra A and let {An} be an increasing sequence of finite type I subfactors in A such that [J™=1Ancz A(S) and (J £L l An is dense in A; then can we choose a bounded perturbation So such that

(S + So)( 0 An)^ 0 An \n=l

J

n=l

and

( j An cz A(S + S0)l n=l

Changing the subject, let S be a generator in a UHF algebra A. For each aeA(S), there is a positive number r(a) such that

Define oo fin/

(\z\
a(=a*)eA(d)

Remark The problem has a negative answer in the general case as mentioned before (cf. §3.4). 4.5.18 Problem If Problem 4.5.17 has a negative answer, under what conditions can one conclude {oez} preserves the spectrum of A(3)l

4.5 UHF algebras and normal *-derivations

167

4.5.19 Problem Can we choose an increasing sequence {An} of finite type I subfactors in A(S) such that there exists a positive number r0for which [J™=iAn is dense in A and £ „ % ( || 8\a) W)tn < + oo (0 s£ t < r0) for ae\J?=1AJ The proof of 4.5.1 does not guarantee the existence of an increasing sequence {An} as the above. On the other hand, in quantum lattice models, we can often choose such a fixed number [159]. In Theorem 4.5.1, we have shown that for a C*-dynamics {si9exp(t8)(tEM)} with a UHF algebra si, there is an increasing sequence of finite type I subfactors {sin} such that les/x c s/2 c • • • c si n c ~-9{J™=1sin is dense in si is analytic with respect to S. Furthermore, by and every element of [j^=1^n Proposition 4.5.5, there is a bounded *-derivation 8ih such that (d 4- 8ih)@ c Q)9 where Q) = \]^^nIt is clear that every element of ^ is a C°°-vector with respect to 3 + 8ih9 but it is not necessarily analytic with respect to 5 + Sih9 because bounded perturbations cannot keep their analyticity in general. If one can choose a 8ih such that every element of 3) is analytic with respect to S + 5ih9 then the C*-dynamics {stf,exp(t(S + Sih))(teU)} is approximately inner, because Q) is dense in $0 and every element of 3) is analytic with respect to the restriction of S + 8ih to Q), for (S + 8ih)@ a Q) (3.4.5). Therefore we can conclude that the C*»dynamics {J/,exp(td)(teR)} is approximately inner. Finally we shall prove the following proposition. 4.5.20 Proposition If one can choose an analytic element with respect to 6 as h in the above consideration, then every element of Q) is analytic with respect to 8 + Sih. Proof Let T be the tracial state on s&9 and let {7iT, Ux,JfT} be the covariant representation of {stf,exp(td)(teM)} constructed via T. Since si is simple, the representation nx of si is faithful. Put at — exp(td) and let Ux(t) = exp(itHx). Then by the theory of semi-groups (cf. 1.17, and §4.4), exp iz{Hx + nz(h)) exp( - izHz)

Since heA{8\ there is a positive number r0 such that nx(<xz(h)) is analytic on Dro° and continuous on D ro , where Dro is the closed disk with a radius r0 at the center 0 in the complex plane and Dro° is the interior of Dro. 1 P~O

p\

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Operator algebras in dynamical systems

where M=sup||7r T (a z (/z))||. Hence z-• exp iz(/JT + 7rT(/i))exp( — izHz) is analytic on Dro° and continuous on Dro. For 7rT(exp z(S + Sih)(a)) = exp iz(Hz + 7rT(/z))7rT(a) exp - iz(i/T + nz(h)) = expiz(// T + 7rT(/i))exp(— iz//T)-expiz/fT7iT(a)exp(— izifT) x exp iz/f T exp — iz(HT = exp iz(ifT + 7iT(/z)) exp — izHxnz(az(a)) exp iz// t exp — iz(if T + nz(h)) Since a e ^ , there is a positive number r2 such that z->7iT(az(a)) is analytic on Dri°. Put r = m i n ^ , ^ ) ; then, z-• ;rT(exp z(d + Sih)(a)) is analytic on Dr°. Since 7rT is an isometry, exp z(S + Sih)(a) is analytic on Dr°, and so a is analytic with respect to S + (5ifr This completes the proof. • 4.5.21 Notes and remarks Theorem 4.5.1 was first proved by the author [166] in 1974. Except for the proof of analyticity the technique used in the proof was first invented by Glimm [207] for the study of UHF algebras. The Powers-Sakai conjecture has been studied by many researchers since the beginning of the theory of unbounded *-derivations in 1974. But, as yet, we do not have any definitive result on it. Proposition 4.5.11 was essentially due to Jaffe and Glimm [215]. However the present form was proved by Herman [76]. References [146], [166], [170], [175].

4.6 Commutative normal *-derivations in UHF algebras A classical lattice system is usually treated in a commutative algebra of functions. However it is more convenient to deal with it as a special case of quantum lattice systems, because we can then define the time automorphism group, and the Gibbs states can be computed as KMS states with respect to this time automorphism group. A commutative normal *-derivation in a UHF algebra is a generalization of a classical lattice system and they have better properties than a non-commutative one. In this section, we shall establish the basic properties of commutative derivations. Let S be a commutative normal *-derivation in a UHF-algebra A - i.e. one can choose a sequence {hn} of self-adjoint elements in A such that 8(a) = i[hn, a] (asAn; n = 1,2,...,), hmhn = hnhm (m, n = 1,2,...,). Let Bn be the

4.6 Commutative normal *-derivations in UHF algebras

169

C*-subalgebra of A generated by An and huh2,..-,hn; then by 4.1.11 there is a unique generator (5X such that S<^S1 and exp(td^(b) = exp(tdihn)(b) (beBn;n= 1,2,...,). Let Cn be the C*-subalgebra of A generated by An and hn; then ^ ( Q c C , , and C n c 5 n , so that exp(tdl)(c) = exp(tSihn)(c) (ceCn;n= 1,2,...,). Now let \j/fi be a KMS state for {^exp^Xfe/*)} at p. Then,

(aeA,beBn).

^ j a exp( - R ) = ^ a exp( - R ) exp(R (note Qxp(Phn)GBn); therefore il/p(ka) = ij/p(ak) (asA\ where /c is any element of the C*-subalgebra of A generated by exp(phn). Define /^p (exp(R)) then (j> is a tracial state on £„, and ^(6) = 0(ft exp( - R ) Now let Qn be the set of all tracial states on Bn and let <3(Qn) be the set of all extreme points in Qn (i.e. all factorial, tracial states on Bn); there is a unique probability Radon measure \in on Qn such that /in(d(Qn)) = 1 and

W ) -

Hence we have



(fteBJ.

Define

then a(bexP(-phn)) Therefore we have the following theorem.

(beBn).

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Operator algebras in dynamical systems

4.6.1 Theorem Let \j/fi be a KMS state for {^,exp(^1)(te[R)} at /?; then there is a unique sequence {vn} of probability Radon measures on the compact spaces Qn consisting of all tracial states on Bn with vn(d(Cln)) = 1 such that

f

«xoo

Moreover if{hn} is another sequence of self-adjoint elements in Bn such that ihn(b) = 3iJ,n(b)(beBn),then o(bexp(-phn)) (7(exp(- /HO)

a{btxp{-fhn)) a(exp(-phj)

{beBn,ae8(nn))

and so vn does not depend on a special choice of(hn). Quite similarly, there is a unique sequence {£„} ofprobability Radon measures on the compact space Fn consisting of all tracial states on Cn with t;n(d(Tn)) = 1 such that

a(Fn)


Moreover if{hn} is another sequence of self-adjoint elements in Cn such that Zn% then cr(exp( — fihn)) o"(exp( — /?/in))

(ceCn,aed(Tn))

and so £n does not depend on a special choice of(hn). Proof It is enough to show that •Phn))

(j(bexp(-phn))

Since a is factorial, a is multiplicative on the center of Bn and since exp( — fi(hn — hn)) belongs to the center of Bn, c(b exp( - /WO) = a{b exp( - fihn)txp{ - P(hn - hn))) = a{b exp( - phn) hence o(bcxp(-phn)) <j(exp(-phn))


(beBn,aed(Qn)).

•

4.6.2 Proposition Let 3 be a commutative normal *-derivation in a UHF 6(a) = i[/jn,a] (ae/4 B ;n= 1,2,...,) algebra A such that S>(S)=\J^=lAn, hmhn = hnhm (m,n=l,2,...,) and hne{J%=1Am (n= 1,2,...,). Then for each

4.6 Commutative normal *-derivations in UHF algebras

171

KMS state 4>p for {A^QxpitS^} at P there is a unique sequence {hn} of self-adjoint elements in Bn such that Sif;n(a) = S1(a) (aeBn;n = 1,2,...,) and
- n = 1,2,...,) and <^(a) = x{aexp(-R))

(aeCn, n = 1,2,...,).

Proo/ Since ^ n e(J* = 1 A m (n = 1,2,...,), Bn is finite-dimensional; hence £„ = XJ= iBnPnj9 where the pn 7- are the minimal central projection of Bn. Since Bnpn j is a full matrix algebra and is invariant under exp(t(51), by the unicity of KMS state at ft MP"j) = T(flPn,jexp( - jg/i> wJ ) = r(aexp( - phn)pnJ) p(Pn,j)

*(exp( - Phn)pnJ)

i(exp( -

Phn)pnj)

Put

-phn)pj then (5i^(a) = (51(a)(aG5J. Moreover

I (aeBn).

D

Let (5 be a commutative normal *-derivation in A such that 6(a) = \[hn, a~\ (aeAn; n = 1,2,...,), hmhrt = /in/zm (m, n, = 1,2,...,). Now suppose that there exists a commutative C*-subalgebra C of A such (n= 1,2,...,), where A'n that hneC(n= 1,2,...,) and C = (AnnC)®{A'nnC) is the commutant of An in A (n = 1,2,...,). (Note: all classical lattice systems and Ising models satisfy this property.) Then Bn = An®(BnnA'n) = An®Zn, where Zn is the center of Bn, for Bn is the C*-subalgebra of A generated by An and C = An®{A'nc\C). Let Zn = C(Kn); then /!„ ® Z n = C(An, Kn) = the C*-algebra of all ^ n-valued continuous functions on the compact space Kn. For aeC(An, Kn), define <&n{a)(t) = i(a(0) (teKn); then On(a)eZn. The
172

Operator algebras in dynamical systems

(1) (2) (3) For teKm a\-^><&(a)(i) is a factorial, tracial state on Bn; moreover if tut2eKn and t1^t2, then they define different states; hence we can consider Kn c d(QJ. On the other hand, for aed(Qn), o is multiplicative on Zn and so there is a point t in Kn such that o{a) = 0>(a)(t) for «eZ n . Since /!„ is a full matrix algebra, it has a unique tracial state and so a(a) = $>(a)(t) (aeBn). Therefore Kn = d(Qn). Hence we have tne following theorem. 4.6.3 Theorem Suppose that S is a normal commutative ^-derivation in A such that 3(a) = \\hn, a] (aeAn; n = 1,2,...,), hmhn = hnhm (m, n = 1,2,...,). Moreover suppose that there is a commutative C*-algebra C of A such that hneC (n= 1,2,...,). Let Zn be the center (n= 1,2,...,) and C = {AnnC)®(A'nnC) ofBn and let Zn = C(Kn). Then for any KMS state xj/pfor { ^ e x p ^ ) } there is a unique sequence {fin} of probability Radon measures on Kn such that

f Quite similarly, one can formulate the corresponding result for {Cn}. 4.6.4 Proposition Under the assumptions of 4.6.3, for any KMS state (j)pfor {A,exp(^ 1 )} at P there is a sequence {hn} of self-adjoint elements in A such that ^(a) = i [ / J n , a ] ( a G ^ n ; n = l , 2 , . . . , ) , / 5 M G C n ( n = l , 2 , . . . , ) and

n -oo

Conversely for any sequence {hn} satisfying the conditions (1) i[hn,al = d(a) (2)

(aeAn;n=l,2,...);

hneCn(n=h2,...),

define < ^ » - T(X exp( - R))/T(exp( - fhn)) (xeA); then any accumulation point of {£,„} in the state space of A is a KMS state for {^exp^i} at p. Proof By a slight modification of the proof of 4.5.5, one can easily show that for any positive number s > 0 there is a self-adjoint element d in C such that {3-Sid) (3>{8)) (§) and ||<5id|| <e. Therefore by 4.6.2, for any KMS state \j/p for {A,Qxp(t(S — 5id))} at f$ there is a sequence {kn} of self-adjoint

4.6 Commutative normal *-derivations in UHF algebras

173

elements in C such that (8 — Sld)(a) = i[/cn, a] (aeAn; n = 1,2,...,) and T(flexp(-j8fcB))

l n ;w=l,2,...).

In particular,

Since {exp(t(^1 — Sid))}(c) = c for ceC, the corresponding KMS state for } at 0 is (^^xexp( - mW^Vi ~ M))) {xeA) (4.4.7); hence x exp( - fid))

Hm

T(xexp(jgfe + J)))

where i[/cn + d, a] = S(a) (asAn; n = 1,2,...). The rest of the theorem is clear.

• Let (j> be a KMS state for {^exp^XteR)} at /?, where d1{a) = i[hma] (aeAn;n= 1,2,...) and {hn} is a mutually commuting family. Consider the covariant representation {n^ U^Jf^} of {A^xpitd^teU)} constructed via = n+(hj (n = 1,2,...). 0. Since (exp^JX/O = hw we obtain U^n^U^t)* Let D be the W*-subalgebra of Ji{ = n^A)") such that D = {aeJt\U+{t)aU+{t)* = a Then for heD,

For

(teR)}.

174

Operator algebras in dynamical systems

hence (xexp(-jWi/2)l,,exp(- /Wi/2)1,) = (xexp( 4.6.5 Proposition Let 4> be a KMS state for {A,exp(f(51)(fG(R)} at /?, with 8 x(a) = \[hm a~\ (aeAn;n = 1,2,...) and a mutually commuting family {hn} (hneAs). Let 82 be another commutative normal *-derivation in A such that ^(^2)= U«°=i^»' ^2(a) — iC^ma] {aeAn;n = 1,2,...) anrf {kn} is a mutually commuting family of self-adjoint elements in A. Suppose that hnkm — kmhn (m, n = 1,2,...). / / (1 jf — in^kn))'1 - > ( 1 ^ — iif)" 1 strongly in Jf^ and l 0 e^(exp( — PH/2)) (in particular, limsupc/)(exp( — flkn)) < + 00), where H is a self-adjoint operator in J«f ^, then

-w.

is a KMS statefor {A, expW^ + S2))(teU)} at & where (S1 + S2)(a) = \[hn + kn, a] for ae@n (<3)n is the C*-subalgebra of A generated by An and /z 1 ? /z 2 ,...,/i n ,/c 1 ,/c 2 ,...,/c n ).

Proof Since (1 -\H)~l£j(,HnJ(

and so

2 Let H = ^^dE(X)

/

v

r

V 2

and Hn = \n_nXdE(X) and put

g(x) = / xexp(

jl^,exp( jlexp(

then || /(i^, /f„) - ^ || -^ 0 (n -> oo). (ijj, H.X^(a) exp( -

= /(i/?,H^n^n^a))

(a,be Q ^ A

Hence <7(7i» exp( - )8(H^ + H))n+Q>) exp(^(H^ + if))) = g(n+(b)K+(a)). 4.6.6 Notes and remarks The study of commutative normal *-derivations is important, because they include classical lattice systems and Ising models.

4.7 Phase transitions

175

Moreover they are much more manageable than non-commutative ones, because they can be treated purely algebraically (cf. 4.1.10). References [167], [168], [169].

4.7 Phase transitions In this section, we shall prove the uniqueness theorem for general normal *-derivations with bounded surface energy. For such derivations we establish the absence of phase transitions, i.e. the uniqueness of /MCMS states for all p. 4.7.1 Definition Let {A, a} be a C*-dynamics and let p be a real number. Suppose that {A, a} has at least one KMS state at the inverse temperature p. Then {A, a} is said to have phase transition at P if it has at least two KMS states at p. If {A, a} has only one KMS state at jS, then it is said to have no phase transition at p. In mathematical physics, we are often concerned with a C*-algebra A containing the identity and an increasing sequence of C*-subalgebras {An} of A such that leAn and the uniform closure of \J™=1An is A. In addition, we are given a *-derivation S in s/ satisfying the following conditions: (2) there is a sequence of self-adjoint elements {hn} in A such that 3(a) = i[/in, a] (aeAn; n = 1,2,...). Such a *-derivation is called a general normal *-derivation in A. 4.7.2 Definition A general normal *-derivation 5 is said to have bounded surface energy if there is a sequence {kn} of self-adjoint elements in A such that kneAn (n = 1,2,...) and \\hn-kn\\= O(l) (n = 1,2,...).

If 6 has bounded surface energy, then by 4.1.9, S is a pre-generator and Qxp(tS) = strong limexp(r<5i/ln). Now we shall prove the following theorem. 4.7.3 Theorem Suppose that An (n= 1,2,...,) has a unique tracial state xn {consequently, A has a unique tracial state T). If a general normal * -derivation 3 in A has bounded surface energy, then the C*-dynamics {A,exp{td){teU)} has a unique KMS state at P for each real number P {namely it has no phase transition at P for each real number P).

176

Operator algebras in dynamical systems

Proof Since exp(td) = strong lim Qxp(tSihJ and A has a tracial state, by 4.3.17, it has a KMS state for each /?. Let (/>l9 v Let 5n = d + Si{kn_hn); then exp(tdn)(a) = exp(tdikn)(a) for aeAn. Since An has a unique tracial state, by 4.4.7 /(ij8, TT^/C,, - ^)X^(a)) = /(i^7r 0 1 (/c w -^))(l^)

T(O exp(

- jSfcJ)

T(exp(-^J)

Since ||7r^(/c n -/zJ||=O(l), by 4.4.7, {/(ijB,rc,(fcn- fcj)} is relatively o{J(%, */#)-compact in JtJ^Ji = n^^A)"), so that by Eberlein's theorem there is a subsequence {/(ijS, n^(kni - hn))} of {/(i/?, n^K - hn))} which converges to a normal faithful state \\i in o(Jt^ Jt\ Hence j K ^

*qe*p(-/M

Quite similarly, we start with 025 then there is a normal state t, on 7fy such that -^^-

(aeA).

Hence \\i(ir . (nW

HIT, (nW

(aeA) and so n4>l is quasi-equivalent to n^ so that (/>! = (j>2 (4.3.12).

D

4.7.4 Proposition Let {A,exp(td)(teM)} be a C*-dynamics. Suppose that {A,exp(td)(teU)} has a KMS state <\> at /?, and there is a sequence {An} of C*-subalgebras in A and a sequence {<$„} of *-derivations in A such that )=>6n and @(5Jn@(5) is dense in A, \\Sn-5\\-=O{l) on \ dn(An) c An9 An has a unique tracial state xn and \)™=1An is dense in A; then {A,exp(t5)(teU)} has no phase transition at jS. Proof Without loss of generality, we may assume that 0 is factorial. Let {?£
/(ij8, /n)//(ij5, /J(l jrj is a KMS state at j? for the dynamics {A, exp(t(<5 + 3iln))}.

4.7 Phase transitions

177

Put 3n = S + 5uj then Sn a Sn and so <5n04M) c= ^4n; hence there is an self-adjoint element kn in n^AJ' such that n^dj^a)) = i[kn, 7i^(a)] (ae^ n ). Since ,4,, has a unique tracial state TW, r0(g))T(aexp(-R)) ^

Tn(exp(-jS/cn))

By 4.4.7, there is a normal faithful positive linear functional \j/ on (4)" = Jl and a subsequence («,.) of (n) such that/(i/?, /nj) -• \\J in o{Jl *, Jt\ Hence

If ! is another factorial KMS state at p for {A,exp(td)(teR)}9 then, by a similar discussion, there is a state il/1 on A which is quasi-equivalent to 4>x and ^1(a)lim Tm.(exp(-jS/cm.)) where (m^) is a subsequence of (rij). Hence 0 and (j>1 are quasi-equivalent and so (/> = (/>!• D mi

4.7.5 Proposition Let {^expfa^XteR)} (resp. {A,exp(t52XtelR)}) be a C*-dynamics and suppose that there is a sequence {dn(n = 3,4,...)} of bounded ^-derivations on A such that {1 — (S2 + ^ J } " 1 ->(1 — ^ i ) " 1 strongly and || <5n || = O(l) {n = 3,4,...). 77zen (/*{4, exp(^ 2 XteR)} /ias a KMS state at p and has no phase transition, then {A,Qxp(td^(teU)} has also no phase transition at p. This proposition is a corollary of Proposition 4.4.11. Let 0 be a KMS state for a C*-dynamics {A,exp(td)(teM)} at P and let {n¥ U^Jf^} be the covariant representation of {A,Qxp(td)(teU)} constructed via 0. Let Jl be the weak closure of n^A) in j f ^. 4.7.6 Proposition Let {hn} be a sequence of self-adjoint elements in A such that there is a dense *-subalgebra @0 of@(S) with (1 - d)@0 = A and || [hn,a~] || ->0 for each ae@0. Define ^ n (a)=/(ij8,7c 0 (fcJXM fl )V/( i i 8 'M'I n)X 1 jr^ then any accumulation point \\J of{\l/n} in the state space of A is again a KMS state for {A,Qxp(td)(teU)} at p. Proof Let 8n = S + ^iftn; then for ae@0, Sn(a) = 3(a) + i[/in, a] -• S(a). Hence (1 _5 f | )- 1 ->(l — 5)" 1 (strongly). By 4.4.7, \//n is a KMS state for {A,Qxp(tSn)(teU)} at j8 and by the proof of 4.3.17, $ is a KMS state for R)}at j8. D

178

Operator algebras in dynamical systems

Let kn = hn + r

l

logf(ip, 7i#n))(l ^ ) 1 ; then

/(.ft »#. Moreover || [kn,d] || -•() for ae@0. Therefore without loss of generality, we may assume that /(ij8,7c#,,))(ljf,)= 1. Put TT^/ZJ =/„; then exp(it(H0 + ZJ) exp( — itif^)G 71^,(^4). For simplicity, we shall assume that (j> is factorial. First we shall note the following fact. || [exp(it(ff, + /„)) e x p ( - itH+\ ^ ( a ) ] || = ||exp(- i*H0)7u4a)exp(itH0) - e x p ( - ir(H0 + Zn))7fy(a)exp(it(tf^ + /„)) || = ||7c0(exp( - t3(a))) - 7c,(exp( -r(<S + ^JXa)) II ->0

for aeA. Now we shall assume that A is separable. Let n^A)" = M and let Ji\ be the pre-dual of M\ then Ji^ is a separable Banach space and so I}(R,J(J* = L°°(IR,^0 (cf. [165]). Let ^(O = exp(it(/f0 + / n ))exp(-it/f >!> = S-'a>dt. Let U 0 = M^) ( ^ ^ , t e R ) ; then and

f/n))exp(-it^),7r4fl)]||||*)lldt. J - 00

Since and „)) exp( - itHJ, n^d)] \\ -»0, = 0 for all i/el^R, J(J. Hence K, <^a](0 = KW, ^ ( a ) ] = 0 a.e. Since ^ is separable, £(t)eJf'nJ? a.e. and so there is a bounded measurable function 1 on R such that £(f) = 2{r)l ^ (reR). Now let Ax (xeJf) be the set of elements (p, q) in L°°(R) © L°°(R) such that || p \\, \\ q || < || x \\. Let A = YlxeJ?K be the weak infinite product space of {\x\xeJi}; then A is a compact space. Let Pnj, x(t) = f(t,hHjXx) and qnj, x(t) = f(t + ip, hn,)(x) (teU); then

^

^

+ g)exp(- it

4.7 Phase transitions

179

and \f(t + i0A,)MI = \f(ip, hmi))(exp(it{H+ + /n,))(exp( - itHJx)

Hence {{paj^qnjJ\xeJ?}eA: Let & = [_{{pn.x,qnjx)\xeJl}\i=\,2,...-\

and let {{px,qx)\xeJ?} be an

accumulation point of J^ in A. Then for each xeJf, there is a subsequence (nk) of (n,) such that ?„,,,-> p x and < 7 n k , x ^ in ^ " ( R ) , L\U)); hence •ip,hj(x)dt :i(t,z)px(t)dt+

I

K2(t,z)qx(t)dt

(say F for

Therefore, F(z)(x) is a bounded analytic function on Sp°. Since f(z, hnk) is a linear functional on Jf, x\-^F(z)(x) is a linear functional on M for each zeSp°. For gel}(R\ let ?/3,x(0 = g(t)Lx
and so

/n.)exp( -

On the other hand,

-

oo

Hence px(t)g(t)dt = and so px(t) = A(0(xl^, 1^) a.e. Now we shall define F(t){x) = px{t) and F(t + i^)(x) = qx(t) {teU); then = k(t)4>{x) a.e, and

180

Operator algebras in dynamical systems

If 0(x) = 0, then F(t)(x) = 0 and so it is continuous. Hence F(z)(x) = 0, so that there is a function /i(z) on Sp° such that F(z)(x) = /i(z)0(x) for xeJi. Since (exp(it(H, + g ) e x p ( - itHJl^

1,) = (exp(k(ff, + lHJ))l^ 1*) = /(*,

is a positive definite function on IR and since f(t,hnj)(l #>)^>k(t) with respect to o{L™(U\l}(U)\ X is a measurable positive definite function on IR. By the well-known theorem, any measurable positive definite function on IR is equivalent to a continuous positive definite function; hence one may assume that X(t) is continuous on IR. Therefore if we define fi(t) = X(t\ \i is continuous on {z|0 ^ Im(z) < )8}. For heA2(S)9 f(t + ijff, 7E,

J

^

+ ;r,(fe)))exp( -

and so generally / ( t + ij?,/!„,

Therefore f(t + ip,hn.) (\#>) is a positive definite function with respect to t. Since f(t + i]8, /znj.)(l ^ ) - F(r + ijS)(l ^ in ^ " ( R ) , L^R)), F(r + i^)(l*} is a measurable positive definite function and so it is equivalent to a continuous positive definite function. Therefore we may assume that t\-^F(t + i/?)(l^) is continuous. Put fx(t + ijS) = F(£ + ij8)(l #>)', then // is a bounded continuous function on Sp and is analytic in the interior Sp°. Moreover we have the following F(z)(x) = I

K,(t, zMt) At Wx) + I

K2(f, z)//(r + i]5) dr U(x)

for 4.7.7 Theorem Let {A,exp(tS)(teU)} be a C*-dynamics with a separable C*-algebra A and let 0 be a factorial KMS state for {A,exp{t5)(teM)} at /?. Let {hn} be a sequence of self-adjoint elements in A such that there is a dense *-subalgebra 90 of 2(5) with (1 - 8)20 = A and || [hn, a] \\ -^0 (ae@0), and put ^n(fl)= /O)8,7t 0(UXM fl))//(i i 8' 7r 4'I »)X1jr^) ( aG ^)- Then any accumulation point i// of{ij/n} in the state space of A is again a KMS state for {A, Qxp(tS)(tG IR)} at p. Moreover if \jj is disjoint with 0 (in particular if \j/ is a different factorial

4.7 Phase transitions

181

KMS state for {A,exp(td)(teU)} at /?), then the sequence

of unitary elements in n^A) converges to zero with respect to

G(U°(U,

Ji\

Proof It is enough to prove that

converges to zero with respect to o{U°{U,Ji\ 1/(11 If this is not true, then one can choose a k such that k =£ 0. Then {i ^ 0, and so \fi(t + ij8)| < /*(ij8) ^ 0. On the other hand, for a(3* 0)eA,

= \f(ip9 /B)(exp(it(H^ + / J ) e x p ( - k ^ /(i/J, ZB)(exp(if(if0 + / J ) e x p ( - i

fa, (exp( -

M

Hence, (-

t5)(a))ll2iKa)1/2.

Since «A is a KMS state for {A,exp(td)(teR)} at j8, |/i(f + ijS)c/>(a)| < ij/(a)1/2il/(a)1/2 = [//(a). Since /i(i/?) > 0,0 is not disjoint with ij/, a contradiction.

D 4.7.8 Proposition Let {A,exp(td)(teU)} be a C*-dynamics with a separable C*-algebra A and let <> / be a factorial KMS state for {X,exp(t<5)(te[R)} at j8. Let {hn} be a sequence of self-adjoint elements in A such that there is a dense *-subalgebra @0 of9>(8) with (\-8)9>0 = A and \\ [hB, a] || -•0 (ae2fo\ and let ^ ( a ) = /(ij?,^(/z n ))(^(a))//(ijS,^(/z n ))(l^) (aeA). Then if a sequence converges pointwise {/(is,7fy(fen))(lj^)} of continuous functions onO^s^fi to a function g which is not zero at /? and is continuous at ft, then {ij/n} converges to cj) in the state space of A. Proof Let / ( i / J . ^ f o M l ^ - ^ i j S ) . Let kn = then f(iP,n
= liml expl —

182

Operator algebras in dynamical systems

Hence without loss of generality, we may assume that /(i^,7r(/)(/in))(l ^ ) = 1. fi(ip)(f)(a) ^ By the previous considerations, F(z)(x) = ii(z)^(x)(xeJi,zeSp\ il/(a)(a^0,aeA\ where \jj is any accumulation point of {\f/n} in the state space of A. On the other hand, fi(\p) = lim fi(is) = lim lim / ( i s , /*„ .)(1 ^ ) s]fi

s]P

j

= lim g(is) = g(ifi = lim /(ijg, /znj.)(l ^ ) = 1. *!/*

j

Hence 0(a) ^ ^(a), and 0 = ^.

D

4.7.9 Proposition Let 0 fee a KMS state for a C*-dynamics {A,exp{tS)(teU)} at ft, and let {/„} be a sequence of self-adjoint elements of n^A)" = M such that supo^/j |/(is, /„)(/„)! < + oo. Then there is a subsequence {f (isJn.)(l jfj} of a sequence {/(is,/ n )(l^ )} of continuous functions on O ^ s ^ / ? which converges to a continuous function on 0 ^s ^ P uniformly. Proof

For /zev42(<5), /(is, 7c0(fc))(l ^ ) - ( e x p ( - s(H 0 +

and so ^ / ( i s , ^ ( f c ) ) ( l ^ ) = ( - (H+ + ds

n+ih))exp(-

- - (7c,(/i)exp(hence ^ / ( i s , ^(fc))(l ^ ) = - / ( i s , ds

n + +

n^hJM For /e^T, take {n+{ha)} such that M ^ ^ I I M M I I < ll'll and (strongly); then / ( i s , 7c0(fca))(7c0(ha)) - • / ( i s , /)(/) uniformly on 0 < s < ]8. Hence

= /(is, ^ Jo

Hence

/(is, 0(1 *.) ~ /(0,0(l>,) = P - /(ip, 0(0dp Jo

and so

~ (1 jr^) = — /(is> 0(0ds

4.8 Continuous quantum systems

Therefore

if

supo^s<^ |/(is,/ n )(/J| < + oo, then 2

{/fe/„,)(/„,)} suchTha ! f(isjnj)(lnj)^g(is)

183

take

a

subsequence

in (7(^(0,^,^(0,^)); then g(ip)dp

Jo

Jo

and so / ( i s , / n ) ( l j f J — / ( 0 , / n ) ( l jf , ) - •

^(ip)dp.

Jo Since /(0, /„ )(1 ^ ) = 0(1 ^ ) = 1,

• 4.7.10 Corollary Let {A,Qxp(tS)(teU)} be a C*-dynamics with a separable C*-algebra A and let be a factorial KMS state for {A,exp(td)(teR)} at /?. Let {hn} be a sequence of self-adjoint elements in A such that there is a dense *-subalgebra Q)o ofQ){S) with (1 - §)3>0 = A and \\ [hB, fl] || -• 0 (ae®0), and let

If sup0<s in the state 1 <M< + 00

space of A. In particular if {\\hn\\} is bounded, then {ij/n} converges to (j>. Remark If {|| hn \\} is bounded, then by 4.4.11, we can eliminate the assumption of separability from the above theorem. 4.7.11 Notes and Remarks Theorem 4.7.3 was first proved by the author [168] for commutative normal ""-derivations in 1975. Later in the same year, Araki [5] proved it for normal *-derivations in UHF algebra. Kishimoto [104] also gave a simplified proof of Araki's theorem. References [5], [104], [168], [169], [170].

4 8 Continuous quantum systems The study of continuous quantum systems is so far incomplete. In fact, for general interacting models, the time evolution has not yet been constructed. This means that the corresponding *-derivation has not been constructed globally for interacting models. In this section, we shall generalize the notion of a C*-dynamical system to include a fairly wide class of interacting models in continuous quantum systems.

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Operator algebras in dynamical systems

We shall formulate a system of four axioms (l)-(4) which is satisfied by a fairly wide class of interacting models in continuous quantum systems, and show that a globally defined *-derivation exists under these four axioms. Next we shall add a further three axioms (5)-(7) to study the existence of time evolution and KMS states, and to show that the combined seven axioms (l)-(7) assure the existence of time evolution and KMS states. Unfortunately it is not so easy to check axioms (6) and (7) for interacting models in continuous quantum systems. These two axioms are more or less automatic if the model has time evolution and a KMS state. It is a problem for the future to check them for general interacting models. We shall give some applications of these two axioms in some concrete models. For continuous quantum systems, another hopeful method might be to consider dynamical systems on unbounded operator algebras. In the last part of this section, we shall present the systems of five axioms (V)-(5r) to formulate dynamical systems on unbounded operator algebras, which might be applicable to a wide class of interacting models. We shall show that these five axioms (l')-(5f) ensure again the existence of time evolution and KMS states. Here again Axiom (5') is difficult to verify for general interacting models. Both Axioms (7) and (5') comprise the so-called core problem in mathematics which appears in various branches of mathematical physics and is a traditionally difficult problem. In the previous sections, we have formulated the theory of quantum lattice systems within the framework of unbounded *-derivations in C*-algebras. In particular, time evolution has been constructed as a strongly continuous one-parameter group of *-automorphisms on a C*-algebra by integrating a given unbounded *-derivation in the algebra. Consequently, C*-dynamics is well fitted to be an abstraction of quantum lattice systems. On the other hand, for continuous quantum systems it is impossible to construct time evolution as a strongly continuous one-parameter group of *-automorphisms on an appropriate C*-algebra. At best, one may hope to construct time evolution as a a-weakly continuous one-parameter group of *- automorphisms on the weak closure of a C*-algebra on an appropriate Hilbert space. This is the case of the ideal Bose gas. However for general interacting models, time evolution has not been constructed even in this weak sense. As a matter of fact, even the *-derivation has not been constructed globally for interacting models. In this section, we shall generalize the notion of C*-dynamical systems so that our system can include a wide class of interacting models in continuous quantum systems and discuss the existence of time evolution and KMS states. In mathematical physics, we are often concerned with a C*-algebra jtf containing an identity and an increasing sequence {stfn} of C*-subalgebras

4.8 Continuous quantum systems

185

of stf such that l e ^ j c ^ c . - c ^ c . " and the uniform closure of To deal with continuous quantum systems, we shall assume that s/n = ?n), where B(^fn) is the W*-algebra of all bounded operators on a separable Hilbert space J^n. By the separability of JlPn9B(Jfn) is weakly closed in B(Jfm) (cf. [165]) and the o-weak topology of B{jfn) coincides with the a-weak topology of B(J^m) on B(J^n) for m ^ n. A linear mapping S in si is said to be a *-derivation if it satisfies the following conditions: (1) There is an increasing sequence {@n} of *-subalgebras of si containing the identity such that 1 G^1 CZ $)2 cz • • • cz Q)n cz • • •. (2) There is an increasing subsequence {m(n)} of {n} such that 2n cz stfm{n) and the cr-weak closure of Q)n in sim{n) contains sin for each n. (3) 3f(8) ={J™=1@n, where 2(8) is the domain of 8, and 8(ab) = 8{a)b + a8(b) and 8{a*) = d(a)* for a,be9>(8). (4) For each Q)n there is a self-adjoint operator hn in Jf m(n) such that S(a) = i[/in, a] forae@n, where (•) is the closure of (•) on the space Jf m(M). Now we shall give some examples of the above system. For details, we refer the reader to Bratteli and Robinson [42]. Let L2(UV) be the Hilbert space of all square integrable functions on the v-dimensional euclidean space (Rv, and let ^"±(L2(RV)) be the Bose-Fock and Fermi-Fock spaces. Let An = {XGUV\ II x || < w}, where || • || is the euclidean norm of Rv and let L2(An) ±m denote the subspace of L2(Anm) formed by the totally symmetric (plus sign) and totally anti-symmetric (minus sign) functions of m-variables xeAn. Consider the associated Fock space J r ± (L 2 (A n )) = X m ^ 0 ^ 2 (^«)± m w ^ t n 2 L (An), and let sf ±(An) denote the CCR and the CAR algebras respectively. Take the a-weak closure of si + (AB) in B(& ± (L2(An)); then it is B(& ± (L2(An)). Now let A be a bounded open subset of the configuration space 1RV such that OeA and — A = A. Consider a Bose or Fermi gas interaction through a two-body potential O; then » dxdy<X>(x —

y)a*(x)a*(y)a(y)a{x).

Suppose that OeL2(A) and /eL 2 (Aj), where A1 is a bounded open subset of R\ Then

= jdx/(x)j | O(x-x i )U (m+1) (x,x 1) x 2 ,...,xJ

for xj/e^±(L2{W)).

186

Operator algebras in dynamical systems

If xi$Al - A (i = 1,2,..., m), then (\_U^ a ( / ) # ) ( m ) = 0. Hence

We can also easily see that

Let W(/) be the Weyl operator / e L ^ A J ; then

< || , / and i^ (cf. Lemma 6.3.34 in [42]). Therefore [I/*, W^(/)] = [ l / . M + A, W(/)], and [ I / . , W(/)](l + ^ A 1 + A ) " 3 / 2 is bounded. 4.8.1 Lemma [t/ OAi + A,iVA2] = 0 and [ l ^ , iVAJ = 0 , wfere A2 is a bounded open subset of Aj + A. Proof For (/)<m)6L2(A2)±m and ^ " e L ^ A ! + A)nA 2 c )±',

On the other hand,

Hence [ [ / ^ +A,NAl] = 0 and analogously [UO,NA2] = 0.

D

Since [Uo,W(f)](l + iV Al+A )~ 3/2 is bounded on $F±(L2(A1 + A)) it is also bounded on ^±(L2(IRV)). Since 1-4-/V 1

~ri>Ai+A

= } -i- N x

'

iy

-\- N

C\ -\- N

Ai "T" n ( A i + A ) n A i c '

V1 '

J>

\C\ 4- N

Ai A 1 ~

2>

(Ai + A)nAi c /

and so

By 4.8.1, " 3/2 fi + N V1 ^

i>

r3/2i

(Ai+A)nAic/

./v r 3 / 2 a + N i>

Ai/

V1

Ti>

-I

r3/2

(A 1 +A)nAi c ^

\

4.8 Continuous quantum systems

187

and IU9, W(f)(\ + N A l )- 3 ' 2 (l + N ( A i + A ) n A i ,)- 3 ' 2 ]

Analogously,

Now let sty = B(^±(L2(A1)) and let 3)x be a *-subalgebra of B(^ ± (L 2 (A! + A)) generated by " {(1 + N ( A i + A ) n A i C )- 3 / 2 (l + AT Ai r 3/2 W(/)(i + N A l ) " 3 ' 2 (i + ^ A . + A w ^ r ^ l / e C o ^ A i ^ C A i ) } and the identity. Let s/2 = BeJ zr ± (L 2 (A 1 + A); then the a-weak closure of © t contains 2 . because j / ^ A J is weakly ^ A ^ ^ A , ^ dense in B(J r ± (L 2 (A 1 ))), and so it contains (1 + N{M + A)nAiC) 3; hence it contains s/lm Next let - V 2 be the Laplacian and let T = dr(— V2) be the kinetic energy operator. Then for feC0co(A2)nL2(A2), for il/G^±(L2(Uv)) (cf. Lemma 6.3.37 in [42]), where A2 is a bounded open subset of U\ Since [T, W(f)] = [7\ 2 , W(/)] and [T, NAJ = [T Aa, ATAJ = 0,

where N is the number operator on J*+(L2(IRV)) and /z is a real number. Moreover [L/o + T — juAT, a] is bounded on ^ ± (L 2 (R V )). Next, starting from J/ 2 > w e construct a *-subalgebra ^ r 2 using a method similar to the one above, and let Q)2 be a *-subalgebra of j/ 3 (=5(J 2 r ± (L 2 (A 1 + A + A)))) generated by Q)x and ^ 2 ; then clearly,

for b e ^ 2 and [ l / o + T — /iAT,ft] is bounded. Moreover the a-weak closure of Q)2 contains j / 2 Continuing this process, we can construct increasing sequences {stfn} and {@n} such that ^ n <= j / n + 1 and the a-weak closure of <2)n in J2/w + 1 contains s/H. Now define a *-derivation S on (J^° = 1 ^ n as follows.

188

Operator algebras in dynamical systems

, where [•] is the closure of [•]. Then S satisfies the conditions (l)-(3). has a self-adjoint extension for each m, then it satisfies Condition (4). This may be done under suitable conditions on the function O, and a suitable boundary condition on — V2. By the above considerations, we have seen that if an interacting continuous quantum model is defined through a two-body interaction potential with v O G L 2 ( A ) (A, a bounded open subset of U \ there is always a globally defined *-derivation 3. Next we shall formulate the conditions which ensure the existence of time evolution and the KMS state. For continuous quantum systems, we may further assume the following conditions: (5)

phn)^e9{hn)} is dense in jfmin) for 0 > 0 and {£eJfM(n)|aexp(where @(hn) is the domain of hn, and exp(— /5hn) is from the trace class in B(Jfmin)).

Now by Condition (5), we can define a state on <stfm(n) for each n as follows: (t>n(a) = T r ( a e x p ( - R ) ) / T r ( e x p ( - R ) ) for aestfm(n). The following two conditions are difficult to prove for interacting models. However, to assure the existence of time evolution and KMS state, they are more or less necessary, though they might be weakened. (6) There is a locally normal state c/> on stf (the uniform closure of \J^=1^/n on J^±(L2([RV))) such that 0(a) = limn_QO0ll(fl) for ae®{8) and (l>(a*d(a)) = \imn^^n(a*S(a))

for

ae@(5\

By the local normality, we mean that the restriction of <j> to stfn is normal for each n. Let <j) be the state on $0 satisfying Condition (6), and let {TT^,, Jf^,} be the GNS representation of $0 constructed via 0 on a Hilbert space Jf ^ Since <\> is locally normal, the restriction of n^ to s/n is faithful for each n; hence n^ is faithful on s/. Now we shall identify stf with n (7) (1 ± 8)Q)(5) is a-weakly dense in Ji, where M is the a-weak closure of (=^) in Now we shall prove the following theorem. 4.8.2 Theorem Suppose that 6 satisfies conditions (l)-(7). Then under the identification of srf with n^s/) in Jt, 5 is a a-weakly closable ^-derivation in M, and its closure S is the generator of a a-weakly continuous one-parameter group {oct\tGU} of *-automorphisms on Jt. Moreover let ^(x) = (xl(i), 1^) for \ then 4> is a KMS state at fi for the W*-dynamical system {Ji,a}.

4.8 Continuous quantum systems

189

To prove the theorem, we shall provide some considerations. Since the a-weak closure of @(d) n srfm{n) in s/m{n) contains srfn and (/> is locally normal, the cr-weak closure of @(8)ns/m{n) in M contains j / n , for n^s/m(n)) is (j-weakly closed in B(J^^); hence the c-weak closure of Q)(b) in M is M. For *(*(«))« lim ,,<*<«)) = lim Let /zp = Y.f=i^jej w ^ h Tr(^) = 1, where {ej is a family of mutually orthogonal one-dimensional projections; then exp(— /tap) = Z7=i ex P(—M/K/; hence Tr(exp(— f$hp)) = X j i x exp(— /M,-) < + oo for all positive numbers p. Therefore except for a finite number of {AJ, A,- must be positive; hence hp is lower bounded. For a positive j60 with j ? 0 < f t 1^1 ^ e x P(^o^j) f° r j ^Jo> where j 0 is some number. Since Qxp( — (P — po)hp)eT(J^m{p)\hpQxp(— php)e T(34?m(p)), where T(Jf m(p)) is the Banach space of all trace class operators on jem{p). Let V={ZeJrmip)\aexp(-php)Ze@(hp)} for ae®p. For £eK, (i/t p aexp(- /Jfcp) - iahpexp(-Php))£

=

d(a)exp(-0hp)£.

Since K is dense in Jf m(p) and \ahpQxp(— fftip) + (5(a)exp(— j8/ip) is a bounded operator, ihpaexp(— f$hp)\ V = iahpexp(— php) + d(a)exp(— php). Since ihpaexp(— php) is a closed operator, i/zpa exp( - j8fep) = iahp exp( - )8fcp) + S(a) exp( - j8/ip). Hence, Tr(<$(a)exp(- php)) = Ti(ihpaenp(-

php)) - Tr(iafc p exp(- /?V>-

Tr(i/jpa e x p ( - /Hi,)) = Tr(i/jpa exp( - p1 hp) exp( - />2hp)), where jffj., j?2 > 0 and ^ + J?2 = j8. Tr(i/j p aexp(- &fe p )exp(- j92/ip)) = Tr(iexp(- P2hp)-hpaexp(-

p,hp)),

for hpaexp( — Plhp) is bounded. Tr(iexp(-/J 2 /i p )-Viexp(-/J 1 fc p )) = Tr((iexp(-/J2fcp)fcp)a exp( - p, hp), = Tr(i/jp exp( - )52ftp)a exp( - px hp)) = Tr(ia exp( - ^ hp)hp exp( - ^2fcp)) = Tx(mhpcxp{-{p,

+ p2)hp)) =

Tr(iahpcxp(-php)).

Hence Tr(<5(a)exp(- php)) = Tr(ia* p exp(- php)) - T r ( i a ^ e x p ( - php)) = 0. Therefore for ae3>n,4>n(d(a)) = 0. If m^n,as3>n implies aeS>m; hence m{5(a)) = 0. Therefore 0(5(fl)) = 0 for ae\)?ml®n. Hence we have the following lemma.

190

Operator algebras in dynamical systems

4.8.3 Lemma (8(a)) = 0 for all ae2(8). Now define \Ha+ = 5(0)$ for ae2(d); then H is well-defined. In fact, for a9be2(8)9 \(K5{a)b)\ = \(ad(b))\ = \(fc*< hence / / is symmetric. 4.8.4 Lemma H is essentially self-adjoint. Proof It is enough to show that (1 + \H)2(H) is dense in Jf 0 . Since (\+\H)2(H) = {(\± 8)2(8)}+9 it is enough to show that (I ±8)2(8) is a-weakly dense in M. By Condition (7), (1 ± 6)2(6) is o-weakly dense in Jl.

D For a, be2(8) with a* = a, = 5(aft)^ - (ad(b% = S(a)b+; hence 3 (a) = i [//, a]. Since i [if, a] is symmetric and i [H, a] <= i [H, a], i[H,a] c {i[/J,a]}* c {i[H,a]}» = i[H^fl]; hence i [ ^ a ] = i [ f £ a ] for ae®(5). By Corollary 3.2.5b in [41], 5 is a^^T^J-closable. For xeB(Jf^), define p,(x) = exp(itH)xexp(— itH); then ti—>pf is a a-weakly continuous oneparameter group of *-automorphisms on B(J^+). Let 3 be the generator of p; then 8a8 and so daS, where 5" is the weak closure of S. Again by Corollary 3.2.5b in [41], ||(1 -A8)(a)\\ ^\\a\\ for ae2(8) and XeR. Now we shall show that (1 ± 8)2(8) = Jl. Since (1 ± 8)2(8) are a-weakly dense in Jl, for X G J there is a directed set { a j in ^((5) such that (1 + S)(aa) -• x (cr-weakly). (1 + J r H l + 5)(aa) = (1 + 5)" 1 !! + 5)(fl«) = fl«-^(l + S'r'x Ja-weakly), because 5 is the generator. Hence (1 + 8)~1xe2(8) and (1 + 5)(1 + ^ ) - 1 x = x and so (1 + 8)2(8) = M. Analogously (1 — 8)2(8) = Jl. Now by the Hille-Yoshida theorem (cf. Theorem 3.1.10 in [41]), S is the generator of a a ^ , (8(a)) = 0 for xs2(d); hence ^(ar(a)) = (a) for a e ^ . Define Mra^ = ar(a)0; then wf can be uniquely extended to a unitary operator on Jf+ (denoted again by ut). One can easily see that t\->ut is a strongly continuous one-parameter group of unitary operators on ffl+. Let

4.8 Continuous quantum systems

191

ut = Qxp(itK) (one may take H as K); then at(x) = exp(irX)xexp(— itK) for xeJi. 4.8.5 Lemma 4> is a KMS state for {J£,a} at ft Proof It is enough to show that by 4.3.16, - ij3<£(a*8{a)) > $(a*a) log(<£(a*a)/<£(aa*)) For

for

aeQ)n, -ip(a*6(a))

Tr(a*[ife p ,g]exp(-^ p )) Tr(exp(-/%_))

Tr(exp(-jS/ip)) Tr(a*aexp(- j8hp)) /Tr(aa*exp(-j "

Tr(exp(-i8fcp))

°g[

Tr(exp(-/?fcp)) /

for p ^ n, because Tr(xexp(- jS/ip))/Tr(exp(- j8fep))(xej2/m(p)) is a KMS state exp(ithp)xexp(-ithp)(xes/mip)). for {j/ m(p) ,y} at ft where yt(x) = Taking the limit, we have {a*a)/(aa*))9 if 0(a*a) / 0 and (j)(aa*) # 0. If 0(a*a) = 0, then |0(a*(5(a))| < (/>(a*a)1/2(/>(5(a)*)(5(a))1/2 = 0. w log(w/f) is defined to be zero if u = 0, v ^ 0; hence

if 0(a*a) = 0. Next suppose that 4>(a*a) ± 0 and 0(aa*) = 0; then - i^p(a*d(a)) > 0p(a*a)log(0p(a*a)/(/)p(aa*)) implies - i)S0(a*5(a)) > 0(a*a){log 0(a*a) - limlog 0p(aa*)}. Hence -i0(a*5(a))^ + oo. This is impossible; hence 0(aa*) = O implies 0(a*a) = 0. Next for ae^(<5), there is a directed set {aa} in 0(5) such that a a ->a (strongly) and (5(aa)-»(5(a) (strongly), for 8 is the a-weak closure of 3. Hence, - ip<j>(a*8(a))

>

0 ( a * a ) ^

192

Operator algebras in dynamical systems

if ($>{a*a) # 0 and $(aa*) ^ 0. If $(a*a) = 0, then (j)(a*S(a)) = 0 by a discussion similar to one above. If (j)(a*a) ^ 0 and 4>(aa*) = 0, then - ip(l)(a*d(a)) ^ 0(a*a)(log (j)(a*a) - lim log $(a a a a *)) a

^ + 00.

This is impossible; hence (f)(aa*) = 0 implies (j)(aa*) = 0. By the above lemma, we have finished the proof of Theorem 4.8.2. Now we shall discuss some concrete cases. 4.8.6 The ideal Fermi gas In this case, we can discuss our subject quite abstractly. Let J ^ . p f ) be the anti-symmetric Fock space built over the one-particle Hilbert space Jf, and let if be a self-adjoint operator in jf. Then the time evolution a is given as follows. at(x) = r(exp(ittf))xr(exp(- itH))

(xeB(^_(JT))),

where T is the second quantization of unitary operators on Jf. The action on the annihilation and creation operators are at(a(f)) = a(exp(itH)f) and a r (a*(/)) = a*(exp(k//)/). Therefore the time evolution can be expressed as a one-parameter group of *-automorphisms on the canonical anticommutation relation algebra ^ _ p f ) . We shall use the notation from §3.1, Example (6), except for using s4 _ p f ) instead of j/(Jf). It is strongly continuous, because || a f (a(/)) - a(f) || = || a(exp(irif)/ - / ) || = || exp(i*H)/ - / 1 | . Now suppose that H is a finite-rank operator with the range space R(H), and let / l 5 / 2 , • • • , / „ be an orthonormal basis of #(//). Let /i>/2>--->/«>/w + i>---/n+p>---be an orthonormal basis of «^. Then a,W/ n+P )) - a(/ n + p ) and a f (a*(/ n+p )) = a*(/M+p)(p =1,2,...). Since s/-(R(H)) is finite-dimensional, a, is uniformly continuous on j/_(K(if)) so that it is uniformly continuous on j / _ ( J f ) . Let at = Qxp(t8); then ^ is a bounded *-derivation. Since s/_(Jt?) is simple, by 2.5.7 there is a self-adjoint element h in j / _ ( J f ) such that 5(a) = i[M](aej*_(Jf)). On the other hand, j / _ ( J f ) is a UHF algebra; therefore there is a unique tracial state x on srf_(#?). Clearly, a KMS state for {j/_(Jf),a} at j? is = T(aexp(- /Jfc))/T(exp(Moreover by the considerations at the beginning of §4.3, such a KMS state at P is unique. Next, let H be a general self-adjoint operator in Jf; then by Weyl's theorem for arbitrary a( > 0), H can be expressed as H = HE + X£, where if£ is a diagonal self-adjoint operator and KE is of Hilbert-Schmidt

4.8 Continuous quantum systems

193

class with | | K J < e . Hence there exists a sequence {Hn} of finite-rank self-adjoint operators such that \\exp{itHn)f — exp(kif)/|| ->0 (rc->oo) for each / e J f , and so a is approximately inner. By 4.3.17, {«a/_(Jf),a} has a KMS state at /?. Now we shall show that the (j> is unique for {^_(Jf),a} at p. Suppose that \jj be another KMS state for {j*_pf),a} at p. Then by 4.3.4 for / G ^ ( e x p ( - pH)\ and t(a*(f)a(g)) = iKa{g)aifi(a*(J))) = ^(a(g)a*(exp(-

pH)f))

= ^«flf,exp(- jBH)/> 1 - a*(exp(- pH)f)a(g)) (use the anti-commutation relation) = <0,exp(- j8tf)/> - iA(«*(exp(- PH)f)a{g)). Hence, (A(a*((l + e x p ( - PH)f)a{g)) = <^,exp( Let hed? and put / = (1 + e x p ( - p H ) ) ' 1 ^ then / e ^ ( e x p ( - p H ) ) and so il/(a*(h)a(g)) =
fifl*(/i)n aw)=*(

n «•(/«) n «(#>< 7=1

n fl*(/«) 1 1 « ( ^ sk / -f \ 1 f

i=2

/

^

7=1

,) nfl*(/«)n ^^) • i=2

7=1

/

Therefore by linearity and the replacement of / x by (1 +exp( —

ft «*(/*) .ft

»=i

j=i

n «*

i=2

7=1

JtP

Hence by induction, the value of \// on the product of a*(/i)a(^j) can be expressed by the sum of products of two-point functions iA(«*(/i)a(^)). If the numbers of a*(fi) and a(gj) differ, then the corresponding value of \jj is zero. In fact, by the previous discussion, if the numbers are different, then ij/(YlT= i a*(fi)T\nj= i a(Qj)) is expressed by the sum of products of numbers with factors il/(Tl™Iia(fip)) o r JA(n^=Ta(^7,))» therefore it is enough to show that W n " = i f l ( / j ) = 0 ( n = l , 2 , . . . , J because HYYj=^{fj)) =

194

Operator algebras in dynamical systems

Ua(fj) =* UfJU a(fj) =* n«(// J=l

/

V

J=2

/

\j=2

= (-iy'1Ja(Gxp{-PH)fi)U

a

(fj))

= ( - 1 ) 2 1 " - 1 ^ a(exp(-2j6H)/1) j j «

Let if = J^oo/ld£A be the spectral decomposition of//. I f / ^ ( [ e , oo)) (a > 0), then \\exp(-kpH)fi\\2=

|exp(-2/cjSA)d||£,/ 1 || 2 ^0

(k-oo)

(assume 0>O). Hence i/'(ll"=i«(/j)) = 0. If / i e £ ( ( - OO, -e]) (e>0), then exists. Replace / t by expikpH)/^, then

(ft«(/,))• Since exp(kpH)fi - 0 (strongly), ^r(nj_ t «(/;•)) = 0.

Hence

( U a(fj)) = \im^(a(E((-8,8))f1)Y\ a(fj)) \j=l

/

£-0

\

j=2

/

Therefore it is enough to show that iA(O"= i a (//)) = 0 under the assumption ///1=0. Suppose that ||/i || = 1; then a(/1 )a*(/1)a(/ 1) = a(f^). Hence, fl

fl(/i)W(a(/i)fl*(/i)a(/i)

- • ( ^ w ) j r i «•/,)»(/.»)

4.8 Continuous quantum systems f

195

n

ft

because a(/ x ) 2 = 0. Hence we have proved the following theorem. 4.8.7 Theorem Let H be a self-adjoint operator in a Hilbert space and let srf-{3f) be the anti-commutation relation algebra over ffl. Let {srf_($f\a} be the C*-dynamics defined by oct = exp(tSiH); then {«s/_pf), a} has a unique KMS state fi for each ^eU and forf9geJtr. Now we shall discuss Gibbs thermodynamic limit or, more generally, KMS states for non-interacting Fermi systems. Let A be a bounded open subset of the configuration space IRV and define a self-adjoint operator HA in L2(A) as a self-adjoint extension of the Laplacian — V2, where —V2 is defined on all infinitely differentiable functions with support in A. The number of the possible extensions is partially governed by the smoothness of the boundary of A. 4.8.8 Theorem For each bounded open subset A a [Rv, let HA be a self-adjoint extension of —V2 in L 2(A) and let H be the unique self-adjoint extension in. L2(UV) of —V2. Let J / _ ( L 2 ( A ) ) be the anti-commutation relation algebra over L2(A) and srf_(L2(UV)) be the corresponding algebra over L2([RV), and let aA and oc be the one-parameter groups of *-automorphisms O / J / _ ( L 2 ( A ) ) and A J / _ ( L 2 ( K V ) ) such that oct (a(f)) = a(exp(i*ifA )/) and a f (a(/)) = a(exp(iri/)/). Then we have the following. (1) l i m ^ * , || a f (x)-a f (x)|| = 0 for all X G J / _ ( L 2 ( A ) ) and all A c Uv uniformly for t in finite intervals of U, where A' -»oo in the sense that A' eventually contains any given A c Rv. (2) Let 0AfAI be the unique KMS state for {j/_(L 2 (A)),a A ' M } with atA^ = exp(t5 iHA _ i;il ) (fieU) at jSelR such that

where z = exp(j8/i); then lim A ^^0 A > (x) = ^M(x) /or X G J / _ ( L 2 ( A ) ) a// A c= U\ where 0M w ^ unique KMS state for {^_(L 2 (R V )), a^} at with af = exp(tdiH_ifll) such that

(3)

0A,M «w^ 0 M ^ r ^ gauge-invariant

- i.e. (j)Atll(yt(x)) = 0 A ,^(x) for

jce

196

Operator algebras in dynamical systems

Proof Clearly || i f A , / - i f / 1 | - > 0 (A'->oo) for /eL 2 (A)nC 0 °°(A) and GO 2 2 V UA C RV(1 ±itf){L (A)nC 0 (A)} is dense in L (R ). Therefore exp(kffA,)-> exp(kif) (strongly), so that ||a, A (a(/)) — oit(a(f))\\ ->0. From this, one can easily conclude that ||a, A '(x)-a f (x)|| ->0 for xe<^_(L 2 (A)) and A c R v . Define <£A>|is(x) = (j)Atll(ys(x)) for X G J ^ _ ( L 2 ( A ) ) ; then a,Ays = yscctA implies:

Hence cj)AJ is given a KMS state for {^_(L 2 (A)), aA'"} at j8, so that by the unicity, cj>A J* = 0A>|1. Analogously, <^ is gauge-invariant. Moreover

/>

for

The rest is clear.

•

Now we shall discuss the relation between the ideal Fermi gas and our system. Let J^_(L2(1RV)) (resp. J^_(L 2 (A))) be the associated Fermi-Fock space with L2(UV) (resp. L2(A)). Let J/_(L 2 (1R V )) (resp. J / _ ( L 2 ( A ) ) ) be the CAR algebra corresponding to

J M ^ v ) ) ( r e s p . J^_(L 2 (A))). Let AB = {peU*\ \\p\\ < n}\ then stf _(L2(An)) is cr-weakly dense in B(^F_(L2(AJ)) = s/n and stf = the uniform closure of

U

oo

^/

Let 9n be the *-subalgebra of s/n generated by {a(/)|/eC c °°(A)} and the identity operator on J^_(L 2 (A)), where CC°°(A) is the space of all infinitely differentiable functions with compact support in An. Let HAn be a self-adjoint extension of the Laplacian —V2 in L 2 (AJ, and let TAn be the second quantization of HAn on J^_(L 2 (A n )). Let K^An = TAn - fiNAn, where fieU and NAn is the number operator on J^_(L 2 (A n )). Then

![W(/)] =


1)/)

and and so

iPW
for / e C c °°(A .)

il)/)

for >

Let ®(5) = U ^ i ® » a n d define 5M(x) = lim^ G O i[K M i A n ,x]=i[K M f A m ,x] for xe{J™=i@n with some m; then 5M is a *-derivation in J3/. It is clear that the above models satisfy Conditions (l)-(4). If we put a classical boundary condition on An, then HAn is lower bounded and exp(-j6(H A n - fil)) has a finite trace value for p > 0 and fieU (cf. Proposition 5.2.22 in [42]). Then by

4.8 Continuous quantum systems

197

Lemma 4.8.4, one can easily see that the models satisfy all of the Conditions (1H7). 4.8.9 Ideal Bose gas The Bose annihilation and creation operators a(f\ a*(f) are no longer bounded; therefore we have to replace them with Weyl operators {W{f)\feL\W)}. W(f) is a unitary operator on J r + (L2((RV)) and satisfies the commutation relation:

Let J * + (A) be a C*-subalgebra of £(#\(L 2 (A))) generated by{K^(/)|/eL 2 (A)}; then the n. Define on Q) as follows. SJ(x) = lim i[K MtAn ,x] = i L X ^ ^ , ^ ]

for

where n0 is a number depending on x; then <5M is a *-derivation in J / ( = the uniform closure of (J^°=1J3/ + (An)). Then, the model satisfies the Conditions (l)-(4). It is known that exp( — PHAn) is of trace class, and if fi(HAn — / / l ) > 0 (strictly positive) then exp( — jSXM An) is of trace class (cf. Proposition 5.2.27 in [42]). Let 0 ( / ) = a(f) + a*(/)/(2) 1/2 ; then
(cf. Proposition 5.2.3 in [42]).

198

Operator algebras in dynamical systems

Moreover i[XMfAn, * ( / ) ] = ®(i(HAfi -

W(f) = exp(i*(/)),

ti\)f).

Therefore for/6Cc°°(AB),

mjo

m!

x (p + 1) 1/2 1| «Alp) || || / || m " J < + oo

for

Let Z' m = o (i)>i!*(/) m = Wr; then

= Z ^riCK^ where ^ is an element in J^ + (L2(A)). Since K^ A > ( r t eL 2 (A n ) + p , iW; iW(f)K*/J'ip)', hence i X ^ A n ^ r ^ - , £ - i P n / ) ^ , ^ . Since Pi^.^^ -^ W^(/)iA(p) and K^An is closed, W(f)il/{p)e@{K^An) and W(f)tip)e i ^ W 1 = i - i ^ ( / ) ^ , , A > ( p ) ' Therefore if ^e@(K^An\ 2 p @(Kn AJ- Since exp( - jSX^ An) preserves L (An)+ invariant, exp ( - j8X^ An) hence { ^ ^ ^ ^ ( A J ) ! ^ / ) ^ ^ ^ , ^ ) } is L\AJ+*czL2(An)+*>n®(K^ny, dense in ^ + (L2(An)). Since (1 + JV AJ" 1 L 2(A II )^c:L 2 (A II )+ p , ( 1 + i V J " 1 ) ^ . Since [XMtAn, (l + i V J ^ l ^ O , K,,An ^ J j S K ^ J has a dense domain, so that Condition (5) is satisfied. Conditions (6) and (7) may need more careful discussion than for the ideal Fermi gas. However the most important point is to show the existence of time evolution and the KMS state for the ideal Bose gas. This can be done by circumventing complicated discussions. Put 0II(x) = Tr(xexp(-i8XMfAii))/Tr(exp(-i»KMiAfi)), whenever 0B(x) is defined. By a discussion similar to the Fermi case, we can show that for/ 1? 0 if m # /, and n(a*(fl)a*(f2)---a*(fm)a(gi)a(g2)---a(gi)) is a sum of products of two-point functions 0n(a*(/,)a(gfJ)) if m = /. Moreover we can easily see that =

4.8 Continuous quantum systems

199

where z = exp(/fyi) (cf. page 60 in [42]), so that

UW(f)) = exp{ -
= exp{ if z < 1. If z = 1 and An satisfies the Dirichlet boundary condition, then we can show again that 4>n(W(f))^(f>(W(f). It is known that there is a locally normal state $ on s/ such that 4> = (f) on the uniform closure of (J *= l srf + (Aw). Define ocnt(x) = exp(itX MtAn )xexp(-itK^ ?A J and itX^) for where K^ is the second quantization of H — fil. ocnt(x)^at(x) in the strong operator topology. Moreover, for/eL 2 (A n ) and - itKJ = W(cxp(it(H for/eL 2 (R v ).

JLW{g)*m,tW(f)) = 0M( = exp( - Im/2) f <(1 +zexp( — (3Hn))(l — zexp( —

x exp I

4

I • exp(-Im<0,exp(itif)/>/2)

h exp( — jS/f ))(1 — z exp( — { (g + exp(ifH)/)> x exp = 4>{W(g)at{W{f)). Therefore by a discussion similar to that for the proof of 4.3.17, we can easily see that 0 is a KMS state at fi for the dynamical system {s/9a}.

200

Operator algebras in dynamical systems

Now we shall consider Conditions (6) and (7). Put . , ._Tr(xexp(-]8X M f A n )) whenever (pn(x) is defined. Let 5£n be a *-algebra generated by {«*(/), a ( / ) | / e C c ° ° ( A n ) } ; then it is known that {„(*)} converges for each x e i ? m (m = 1,2,...,). Denote its limit by (p(x). Then and

[iKMtAB,

for/GC c °°(A n ), so that Mf AB, « • ( / ) ]

= a*(itf „ / )

and

[i/CMtAn,

Therefore [iK MiAn , x ] = [iX M , x ]

for xeJS?B.

Define ^(x) = lim [iXM An, x ] for x e ( J ® = 1 ^ f m ; then (5 is a *-derivation defined on ( j £ = 1 if m (denoted by i f ) Clearly 0 n (5(x)) = 0 for x e ^ n and so <£(<5(x)) = 0 for x G ( J * = 1 i f m . This n is also implies that Condition (6) is satisfied for the *-algebra [j^=1^ma K M S state at p for {B(J^ + (L 2 (A n ))),a M } with

hence - i/ty n (x*)a(x)) > ( / > n ( x * x ) l o g ^ * ^ 0 n (xx*)

for

By taking a limit, we have ^ ^ (*)

for

Consider the GNS representation {n^, Jf ^} of the *-algebra if constructed via 0. Define iH^x^ = (^(x))c/> for xeif; then, H 0 is a symmetric operator in «#%. Let H 0 be the closure of if^ in ^ . Let at = exp(tS); then S(x) = i[KM, x] for x e i f in J^ + (L2(KV)). (1 - ^ ) " 1 a(fMfi)"-a(M = ftaiexpiitHJfJ a(exp(itHJf2) • • • a(exp(ifif „)/„) exp( - *) dr. Since

- t)dt x \ Jo

4.8 Continuous quantum systems

C 2 | | / 1 | | 2 | | / 2 | | 2 - - - | | / J | 2 e x p ( - ( s + f))dsdt

201

< + oo,

o Jo where C is a suitable constant,

3(a)(exp(itHJfl)a(Qxp(itHJf2)

• • • a(exp(itif )/„)) exp( - t) dt

o

Hence {(1 — 3)~x a(f1)a(f2) - • a(fn)} ^^(H ^ and so (1 —iH^^iH^) is dense in Jf ^. Analogously (1 + ifi^(H#) is dense in J-f ^ so that //^ is selfadjoint. Therefore e x p ( i ^ ) ^ ( / ) e x p ( - itH^) = exp(kK M )^(/)exp(-itK M ) on JT+ for feL2(Rv). From this one can easily conclude that the linear span of {((1 ±3)W(f))(f)\feCccc(Rv)} is dense in jf^ and so {(1+<5)(J* =1 0 B ) 0 } is dense in Jf ^, so that Condition (7) is satisfied. 4.8.10 Interacting models To discuss more general interacting models we shall reformulate our axioms for continuous quantum systems by using unbounded operator *-algebras. Let L2(UV) be the Hilbert space of all square integrable functions on the v-dimensional euclidean space Rv, and let J*+(L2([RV)) be the Bose-Fock space and Fermi-Fock space over L2(UV). Let An = {xeUv\ \\x\\ < n}, where || • || is the norm of the euclidean space Uv, and let L 2 (AJ ± m denote the subspace of L2(Anm) formed by the totally symmetric (plus sign) and totally anti-symmetric (minus sign) functions of m-variables xeAn. Consider the associated Fock space
3(a*)\V± =3(0)* for ae@±, of (•) to V± respectively.

where (*)l^± implies the restriction

202

Operator algebras in dynamical systems

(2')

For each n, there is an ^ ±(L2(Am(n))) with n^m(n) and self-adjoint operator hn in & ±(L2(Am{n))) such that d(a) = i[/in, a] for ae@+(An).

(3')

For P > 0, a exp( — R ) and aS(b) exp( — R ) are of trace class for each n and a, &e® ± (AJ, where (0 is the closure of (•) in ^ ± (L 2 (R V )). Now let be a locally normal state on the C*-algebra j2/ ± (the uniform closure of \J™=1B(^ ±(L2(An))); then there is a unique trace class positive operator Tm{n) on ^±(L2{Am(n))) such that ${x) = Tr(xTm(n)) forxeB(^±(L2(AmJ)).

Since a and aS(a) are closed operators in «^±(L2(Am(w))), we can consider the closed operators aTm(/l) and ad(b)Tm(n). Assume that they are of trace class; then we can define (j)(a) and (j)(aS(a)) by Tv(dTm(n)) and Tr(^)Tm(n)). (4r) There is a locally normal state (j) on s/± such that lim^^^ 0n(a) = <j>(a) and Iimll_oo0ll(fl*5(6)) = 0(fl*5(&)) for a,be@ + , where (j)n(a) = T r ( a e x p ( - R))/Tr(exp( - R ) ) and 0B(a*5(fl)) = Tr(oM(b)exp(-R))/ Tr(exp(- jSftJ) for a,be@±. Now suppose that a *-derivation 5 satisfies Conditions (V)-(4'). By (3'), S(a) exp( - Phn) = i[fcB, a] exp( - /JfcJ = i(fcna - ahn) exp( hence S(a) exp( - )8fcn) =3 ihna exp( - R ) - \ahn exp( - R ) . By a discussion similar to that in the proof of Theorem 4.8.2, ahnexp( — /3hn) is of trace class and so 5(a) exp( - phn) + \dhn exp( - phn) => \hna exp( - jSfcJ; hence S(a) exp( — j8ftn) + \ahn exp( — jS/in) = \hna exp( — j9/in). Therefore, (j>Ma)) = Tr((5(a)exp( - jS/zM))/Tr(exp( - R ) ) = Tr(iMexp( - R ) - ia/znexp( - R ) ) / T r ( e x p ( - R ) ) . Let hn = Xj°= i ^ / j , where dim(^) = 1 and {e}} is a mutually orthogonal family of projections with XJL i^/ = 1- Then,

f

where ^

- R ) ^ , y - f (ia exp(- R)^, ^ )

= ^ with ||^|| = 1. Hence Tr(i/i M aexp(-R)) = Tr(aexp(-R)i/z n ),

4.8 Continuous quantum systems

203

also Tr(ia/znexp( — phn)) = Tr(iaexp(— Phn)hn) and so (t>n{S(a)) = 0 for define Therefore we have cj)(S(a)) = O for ae@±. For xeB{^±(L\Am{n)))\ ^/j(x) = Tr(xexp(-j8/iM))/Tr(exp(-jS/zM)); then \\tn is a KMS state at j? for the W*-dynamical system {B{3? ±(L2(Am(n)))),yB}, where yn,t(x) = for xeB(^±(L2(Am(n)))). Therefore, exp(ithn)xexp(-ithn) $

^ ~ ^ (*)

for

where yn r = exp(tdn). Consider the GNS representation {n^J^^J of B(^±(L2(Am(n)))) on a Hilbert space Jf^n constructed via \j/n. Define \Hnxxj/n = S(x)lj/n for xe@ + ; then H n is an essentially self-adjoint operator in Jf^n for xe<3 + . Let Hn be the closure of Hn; then , «/(exp( -

2 By Condition (3'), for ae^±(Am(M)),<2exp( — flhjl) is of trace class; hence we can define the image dlf/n of a in J-f ^ uniquely. Moreover,

and so we have 8(a\n = i[hn9 a\n = iHJa^

for ae@±(Am{n)).

Therefore, ^

^

for

ae®±(Am(n)).

By taking a limit, we have
for a e ^ ± .

Let {71^,^^} be a GNS representation of srf ± on a Hilbert space constructed via (/>. For x,

= ( x V ^U) = Tr(>;*xTm(n)) = Therefore we can define the images a^ and 8(a)^ of a and (3(a) (aeQ) ± ) in Jf ^. Define i i f ^ = (5(a)(/>; then one can easily see that H^ is a densely defined symmetric operator in J#^.

204

Operator algebras in dynamical systems

Now we shall consider the following condition (5')

{[(1 ±d)a](f)\ae@±} is dense in J ^ . Then the closure of H^ is selfadjoint. Let 7r^(at(x)) = exp(irH^)7r^(x)exp( — itH^) for xestf + ; then the state cp on s/± is a KMS state at ft for the dynamics {«$/ + , a}.

Therefore we have the following theorem. 4.8.11 Theorem If a dynamical system {s/±,3} satisfies Conditions (l')-(5'), then it has a time evolution tt-nxt(teU) and a KMS state at /? (/? > 0). Now we shall apply the above system of conditions to interacting Bose models. Let O be an even positive function on Ux with compact support K which is continuous except possibly at the origin. Consider a two-body interaction

Then,

Let T^ An(1) denote the self-adjoint extension of — V2 corresponding to the Dirichlet^oundary condition ^ = 0 on dAn. Define T0OfAn = dr(TG0>A(1)) (the second quantization of 7^>An(1)). Let HOOtAn = TQ0>An + l/^ An (the sum is not necessarily the operator sum) (cf. page 356 in [42]); then H^ An is self-adjoint. Moreover exp{ — P(HaOtAn — fiNAJ} is of trace class for all / ? > 0 and fieU for Fermi statistics and all \i < oo for Bose statistics (cf. Proposition 6.3.5 in [42]). F o r / e C ^ A J , - y)a*(x)a*(y)a{y)a(x) dx dy,, f/(zja(z)dz"| f(z)a(z) dz J

— y)f(z)[a*(x)a*(y)a(y)a(x\

= ff

a(z)~\ dx dy dz

(D(x - y)f{z)( - 3(x - z)a*(y)a(y)a(y)a(x)) dx dy dz ; - y)j\z)( - d(y - z)a*(x)a(};)fl(x))dxdydZ

- y)f(x)a*(x)a(y)a(x)dxdy -

\\
4.8 Continuous quantum systems

205

Hence

C Quite similarly,

Take a positive integer m(n) such that K + An cz Am(n); then for i[T + L/o - MAT, a] = i C T ^ , + l/o,Am(n) - ^NAw(n), a], where T = dr( - V2) in ^+(L2(RV)). Define S(a) = i[T + l/ o - /xJV, a] for a e ^ + , and put =

Tr(aexp(^(^,A^C/^.iVAJ)) Tr(exp( - « r + [/ /iJVJ))

for

Then, G) An (fl*(/i)fl*(/2)' •' a*(fm)a(gm)a(9m

-1)

= x p A n ( yi9

y2, - • • > ^

^i^

2 ?

• • • ^m)>

where p ^ ^ ^ y 2 ,..., _y m ;x 1,x 2 ,...,x m ) are the Dirichlet reduced density matrices, which are a positive bounded continuous function and a sequence {P\n} converges uniformly to a function p on every compact subset in n in [42]). Therefore, Rvm x Rvm ( cf C o r o i i a r v 6320 lim a) An (fl*(/ 1 )fl*(/ 2 ) • • • a*(fm)a(gm)a(gm _ x m

= lim dxx dx2---dxmdy2-'-dymg1(y1)g2(y2y--gm(yJf1(x1)f2(x2)'-'fJ<xJ n J

=

dXidx

^^

Moreover, it is known that co will define a locally normal state (cf. Theorem 6.3.22 in [42]). Now we shall examine Condition (4'). For/EC c °°(A n ), S(a(f)) = i l T ^ ^ + L/o>Am(n) - ,,JV

w

2

= «(K - V -

206

Operator algebras in dynamical systems

and ».>W,, - A** > w «*(/)] = a*(i( - V2 -

A*1)/).

On the (Dther ]hand,

'(/)] =•i f

[

dxd};;)ra*(x)a*(y)a(>')a(x), \j(z)a{2

J Am(n) J Am(n)

L

J

1 dx dy dz<6(x - y)f(z)\_a*(x)a*(y)a(y)a(x), a{z)~\ n) JA m (,

1 41J,

dx dy
Similarly,

i f

f

/)] = -

i[[/

dx dy4>(x -

4- -

y)f(y)a*(x)a*(y)a(x)

dx dyQ>(x —

y)f(x)a*(x)a*(y)a(y).

J A m (n) J A m (n)

On the other hand,

Am(n)

+ ...

_ 2) • • • a ( 3 l

+co^mJa*(f1)a*(f2)-a*(fm)a(gm)a(gm.1)--S(a(g1)))

Moreover, Am(n)((/i

= i dx1 dx 2 ---dx m dxd>;d>' m _ 1 dy m _ 2 ---dy 1 / 1 (x 1 )/ 2 (x 2 )--/ m (x m ) x ( - d>(x - y)fifm(x))3m- i ( y m x(x1,x2,...,xm,y;y,x,ym_l,...,yl)

x(xl,x2,...,xm,x;x,y>ym_v...,yi) /i)«*(/2) • • •fl*(/Ji

i)-gi{yi)pA + ~ dXi

References

207

Quite similarly, we can show that 00

coAn{a*d(a))->co{a*d{a))

(n-> oo) for ae [j ®+(AB) = @ +

Therefore Condition (4') is satisfied. It is difficult to prove Condition (5') for interacting models and this is one of the most important problems in the continuous quantum system. We shall leave this important problem for future research. In any case, it is one of the most important problems to construct time evolution and KMS states for interacting models. References [41] [42] [177] [178].

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Index

actions by Lie groups 13 actions by Un 27 affiliated 14 algebraic quantum field theory 33 amenable 111 analytic element 74 analytic version 7 approximate identity 54 approximately bounded *-derivation 65 approximately inner C*-dynamics 104 approximately inner *-derivation 62 approximately uniform C*-dynamical system 50 Arveson's spectral theory 25 asymptotically abelian 112 automorphism group 11 automatic continuity 22 a-abelian 112

closure of derivation 17 of operator 11, 14 commutation theorem 5 commutant 14 commutation relation 17 commutative C*-algebra 3 commutative normal *-derivation 161 commutative W*-algebra 3 compact operator 63 complex version 8 concrete C*-algebra 3 conservative derivation 60 continuous quantum system 183 convergence on geometric vectors 11 core problem 163 covariant representation 35, 49, 176, 177 cyclic vector 6, 24

Banach algebra 1 bounded derivation 16, 22 bounded perturbation 130 bounded surface energy 105, 175

definition of derivation 17,56 derivation theorem 34 dissipative 60, 65 domain of derivation 65

C*-Algebra 1 C*-dynamics 103 C*-dynamical system 12 C*-infinite tensor product 56, 102 calculus of representations 12 canonical anti-commutation relation algebra 57 Cantor function 91 ceiling state 117 center of W*-algebra 36 central envelope of projection 38 centrally orthogonal 113, 123 Choquet simplex 123 classical interaction 103 classical lattice system 102 closable derivation 17, 56

effective 95 entire analytic element 76 ergodic 111 everywhere-defined derivation extreme point 6 face 109 factor 6 factorial state 6 faithful state 75 finite range 103 finite range interaction 102 flow 96 forward light cone 31 Fourier transform 69 from representation to state 6

22

Index

218 from state to representation 5 from VF*-dynamical system to C*-dynamical system 48 GCF 103 Gelfand-Naimark theorem 5 general group 31 general normal *-derivation 105 general commutative normal *-derivation 106 generalized Cantor function 91 generator 72 geometric vector 10 geometric element 76 GNS representation 5 ground state 48, 107 group algebra 14 Haag-Kastler-Araki axioms Haar measure 14 Hamiltonian 24 harmonic function 6 Heisenberg model 57

33

ideal Bose gas 197 ideal Fermi gas 192 infinitely differentiate element 14 infinitesimal generator 72 inner derivation 34, 56 inner product (scalar product) 15, 33 interacting model 201 interaction 102 invariant state 49, 51 Ising model 56 isometric 2 isotopy 33 Kaplansky's density theorem KMS state 114,115 Lie group 46 local algebra 33 local commutativity 33 lower boundedness condition

5

negative result 45 non-negative spectrum 24 norm continuous C*-dynamical system 12 normal element 19 normal *-derivation 158 normal linear functional 6 norm topology of automorphism group 12 26

quantum lattice system 102 quasi-free *-derivation 57 quasi-ground state 113 quasi-local W*-algebra 33 quasi-well-behaved *-derivation

7

60

representation theorem for C*-algebra and W*-algebra 5 restricted C*-system 14 scalar product (inner product) 15, 33 self-adjoint element 2 separating vector 44 Silov algebra 80 spectrum condition 24, 27, 28, 30, 33, 48 state 2 state space 6, 117 strong operator topology 4 strong lima n 103 strong convergence of the one-parameter groups of *-automorphisms 103 *-automorphism 2 *-derivation 17 *-homomorphism 2 ""-isomorphism 2 (7-weak topology 2, 4 on automorphism group 12

47

modular involution 122 modular operator 122 multiplier algebra 40

observable 24, 102 one-parameter group outer derivation 34

perturbation expansion theorem perturbation theory 7 phase transition 175 of ground state 113 physical ground state 111 positive element 2 positive energy 33 positive linear functional 2 Poisson kernel for the strip 7 Powers-Sakai conjecture 163 predual 2 pregenerator 72 primitive ideal 20 pure state 6

time automorphism group 101 time evolution 103 Tomita-Takesaki theory 122 total energy 102 trace function 4 tracial state 3 transformation group 94 translation invariant 102 two-body potential 185 type I 33 UHF algebra 103, 155 unbounded *-derivation 55 uniformly continuous C*-dynamical system 12,41 uniformly hyperfinite C*-algebra 103

219

Index uniform topology 3 of automorphism group 12 unitary representation 15 vacuum state 33, 48 von Neumann algebra 1 von Neumann double commutant theorem W*-algebra 1 W*-dynamical system

12

weak topology 2 weak operator topology 4 weakly approximately inner C*-dynamical system 104 weakly closed self-adjoint algebra well-behaved element 60 well-behaved *-derivation 60 well-behaved 8* 73 zero-temperature state

107

4, 5