Math. Nachr. 145 (1990) 169-186

One-Parameter Groups Arising from Red Subspaces of Self-Dual Hilbert W*-Moduli

By MICHAEL FRANK of Leipzig

(Received March 9, 1987)

Abstract. The purpose of this paper is to generalize the results of M. A. RIEFPEL and A. VAN DAELE [8, 30 1, 2,3] for HILBEET W*-moduli over commutative W*-Algebras. Some special real sub- @paces of such HXLBERT W*-moduli and the related operators are investigated. Particularly, the relation is established between *weakly continuous unitary one-parameter groups of operators arising from them and the generalized K.M.S. condition. All key definitions are formulated with- out any commutativity supposition for the underlying W*-algebra. The interpretation of these results is given for sets of continuous sections of “self-dual” locally trivial HILBERT bundles over hyperstonian compact spaces. At the end of this paper some aspects of the general noncommu- tative case are discussed.

8 1 Introduction

A (left) pre-HILBERT A-module over a certain C*-algebra A is an A-module X equipped with an A-valued nondegenerate conjugate-bilinear mapping (a, .): X x X + A, (., .) being A-linear at the first argument. The linear space X is HILBERT if it is complete with respect to the norm 11-11 = [I(-, .)llg2. We suppose always that the linear structures of A and of X are compatible. For basic facts concerning =BERT C*-moduli we refer to [7]. A HILBERT A-module 3e over a C*-algebra A is called selfdual if every bounded module map r : X 3 A is of the form (-, a) for some a E X. In this paper we restrict our attention mainly to HILBERT W*-moduli. For them some more fact$ are known as in the general case. We need the following ones :

Defiiition 1.1 [a, Def. 71. Let A be B W*-algebra, X be a pre-HILBERT A-module and P be the set of all normal d t a t e s on A. The topology induced on X by the semi- norms

f ( ( - , *)P, f E P,

f ( ( . ,Y ) ) , Y E a, f E p9

is denoted by tl. The topology induced on X by the linear function&

is denoted by t2.

170 Math. Nechr. 145 (1990)

We remark that, in general, the topology T? is weaker than the topology tl, and that they both are weaker than the norm topology. Throughout this paper we use the following notation. If X is a subset of the HILBEBT W*-module 2, [X]T denotes the set {h: 1 E R,, z E Xo} where Xo is the t,-completion of the set {z E X: 11z11 5 1).

Theorem 1.2 [4, Th. 91. Let A be a W*-aQebra and 3e be a HILBEBT A-module. The following conditions are equivalent:

(i) 3t' is self-dwcl. (ii) The unit ball of 3e is complete with rmpect to the toplogy tl, i.e. X = [XK. (iii) The unit ball of 3e is crnnplete with respect to the top-y t2.

Corollary 1.3 [4, Cor. 111. If A is a W*+ebra and X is a self-dual -BERT A- module the linear span of the range of the A - v a l d inner product on 3e becames both a. W*-subalgebra and an &-leal in A.

Theorem 1.4 "7, Prop. 3.10.1. Let A be a W*-algebra and X be a self-dual HILBEBT A-module. Then, the set EndA (a) of all bounded A-linear operators on d is a W*-alqebra.

These facts make clear that in the caae of 3t' being a selfdual &BEET W*-module, the spectral theorem ([lo, Th. 1.11.3.1) ie valid for each self-adjoint element of EndA (X). Moreover, there exists a polar decomposition for each element of EndA (a) in EndA (d). This is of importance throughout this paper.

Now let A be a commutative W*-algebra, 3t' be a selfdual HILBEBT A-module. 8 2 deals with real selfdud &BERT Ah-dubm0dda X of X and with some bounded operators arising from them. In particular, a conjugate-A-linear involution J on 3e with respect to X is defined. $ 3 invatigatert the strongly continuous one-parameter group {A" : t E R} defined for each X E 3t' on d. The relation of this group to a gen- eralized K.M.S. condition is established. We remark that the main definitions of s 2 and 3 are formulated without the restriction to A to be commutative. $ 4 gives the interpretation of the results of the former paragraphs in terms of locally trivial HILBERT bundles over hyperstonian compact spaces. In $ 6 mme aepects of the general non- commutative c w are discusged. Other applications can be found in [b] .

On some real subspaces of self-dual Hubert W*-moduli and related operafms

§ 2

Let A be a W*-algebra and 2%' be a aelfdual HILBERT A-module. We guppose without loss of generality that the linear span of the range of the A-valued inner product on X' is identical with A, (cf. Cor. 1.3.). Let X be a norm-closed real rtubspace of X being invariant under the action of g ( A ) h (the self-adjoint part of the centre of A) and having t,he following properties :

(i) X n i X = {O}. (ii) X + i X is norm-dense in X. (iii) X = [XI,.

As an example one can take A = X and Ah = X. The following awrtionrt make clear why the third condition above is necessary.

Frank, One-Parameter Group 171

Definition 2.1.

X I = {x E b: (x, y) + (g,x) = Ofor any y E X } .

Proposition 2.2. X I ZS a nornt-clmed real eub8pace of b being invariant under the

Proof. The connection X n X I = {0} follows from the nondegeneracy of the A- action of J ( A ) h and sa t i s f y i~q the con&itions X n X I = {0), X I = [ X l r .

valued inner product on %. The equality

(a E z(A)h, a E X l , y E X) shows that X1 is a real aubspace of b being invariant under the action of J ( A ) h . If we consider a bounded net (xu : xu E X I , a E I, t,-lim xu = x E a}, we get

0 = lim f((% 9) + (3, xu)) .€I

= f ( b 7 9) + (97 3))

for any y E X, any normal state f on A . That is x E X I and, thus, by Theorem 1.2. X l = [Xl];. The norm-completeness is obvious now.

Corollary 2.3. ( X l ) l = X .

Therefore, we have shown the necmity of the condition (iii) above. We remark that this condition is satisfied automatically if A is finite dimensional, (i.e., particularly, if b is a HILBERT space). Now we pass on to cont3iderations about the properties of such real subspaces X of b supposing A to be commutative.

(Q(47 9) + (97 &(a)) = (87 &(?I)) + (Q(? I )7 x)

= (Q(X)> &(!I)) + (a!!), ax)) for any x, y E X .

Proof. We can consider the real self-dual HILBERT Ah-module X, = (b,(., -) +(., -)*} since A h is a real W*-algebra with identical involution. The get X , = (x, (., .) + (a , a)*}

is a real self-dual HILBERT Ah-gubmodule of b,. From [4, Th. 9, Proof] we draw that b, = X , + X,l. Thus, there exists a projection P: b --f X defined by the rule P: X, + X,. The projection P satisfies the condition (1) by its definition and it is

172 Math. Naohr. 146 (1990)

AJinear. Keeping in mind that P acts on X a0 the identical map the inequality

= (W), P(*)) + ((1 - P ) (4, (1 - P) (2)) 2 (P(x) , P(x) ) Y

being valid for any x E X, 0hOws llPll = 1. The proof is analogous for Q. The projection0 P and Q do not commute, in general. We define R := P + Q and

JT := P - &, where T is a positive self-adjoint operator and J is a partial isometry.

Lemma 2.5. R is an injective A-linear operator on X. The connection 0 5 R 5 2 is

Proof. Denoting by i the square root from (- 1) we can state the equalities iP = Qi valid. The c~clme is true for the operator (2 - R).

and iQ = Pi. Chosing arbitrarily c = c1 + c2i E A and x E X the equality

R(-) = P(-) + &(a) = ci * P(x) + c2 * P ( b ) + ci * Q(x) + c2 . Q(0)

= c1 - P(x) + c2i . Q(x) + c1 - Q(x) + c2i - P ( z )

= c * ( P ( 4 + Q(s)) = c - R(x)

is t~atisfied, i.e. the operator R is A-linear. If now R(x) = 0 for a certain x E X, we obtain

(3) 0 = (R(@,=) + 6, R ( W

= (P(a), a) + (3s P(x)) + (Q(x), x) + (*, &(=I) = 2 (Vw, P ( N + (&(=), QW) 9 cf. (1).

Consequently, P(x ) = Q(x) = 0. Since X + i X is dense in 3t' there follows x = 0. Thus, R is an injective operator. Furthermore, since R has a conjugate operator R* in EndA (X) the equality below is valid :

(4) = (R2@), Y) + (3, R V ) )

= ( (P + PQ + QP + Q ) (x), 3) + (Y, (P + PQ + QP + Q) (3~)) = ( (P + Q ) (s), (P + Q) (Y)) + ( ( P + Q) (Y), (P + Q) (*)), = (R(4 , R(Y)) + @(?I), R(=))

= (R*R(x), 3) + (Y, R*R@))

cf. (1) 9

for any x, y E X. Therefore, (R - R*) R = 0. Because of the injectivity of R we draw R = R*. Uaing (3) we get

@(a), 8) = 1/2 * ((R(*), 8) + (x, R(x)))

= 1/2 * ((P(=), W)) + (Q(Z)Y QW)) 2 0

Frank, One-Paremeter Groups 173

for any x E X. On the other hand

(R(x), 8) 5 2 * (G 2)

for any x E X by ( 2 ) . Consequently, 0 5 R 5 2. The proof for (2 - R ) is analogous changing P, Q to (1 - P), (1 - Q).

Lemma 2.6. ( P - Q), T and J are injective operators. ( P - Q ) and J are conjugate-A- linear, whereas (P -Q)2 and Tare A-linear. The equalities J2 = id% and T = R 1 4 2 - R)ll2 are valid, and the operators J and T commute.

Proof. The following equality is satisfied for any a = a, + a2i E A, (a,, a2 E A,,), and any x E X:

(P - Q ) (aac) = a, - P(x ) + a2 - P(ix) - al - Q(x) - a2 * Q(&) = a, - P(x ) + a2i - Q(x) - a, - &(a) - a2i * P ( x ) = (al - a 2 i ) . ( P - Q ) (x ) .

That is (P - Q) is conjugate-A-linear and, correspondingly, (P - Q ) B is A-linear. Moreover, ( P - &)a = P - PQ - QP + Q = ( 2 - R) R, and thus, (P - Q)2 is a self- adjoint positive, injective operator on X. We define the operator T : X + X by the formula

T = [ ( p - Q)2]1 /2 = (2 - R)1/2 Rl/2.

Thus, T is an A-linear, bounded, injective, self-adjoint positive operator on X. The sets [T(X)]: and [T2(X)]; are both equal to 3. There exists a map J for which the equality JT = (P - Q ) is satisfied on X since the operators T and (P - Q) are injective and T2 = ( P - Q ) 2 . J maps T ( X ) into X. From the equality (P - Q)2 = JTJT = P we draw T = JTJ since T is injective. Hence, J can be extended to a map defined on 3 being bounded, conjugate-A-linear and injective. Furthermore, the existence of the inverse operator J-1 defined on X being bounded, conjugate-A-linear and injective is guaranteed. We obtain JT = TJ-1 and T2 = JTJT = JTBJ-1, i.e. JTB = TZJ. Congequently, J commutes with T2, and therefore, with T. Moreover, J = J-l on X and J2 = idw.

Lemma 2.7. T commutes with P, Q, and R. The eqwlit&s J P = ( 1 - Q ) J , J Q = ( 1 - P ) J a n d J R = ( 2 - R ) J a r e u a l i d .

Proof (cf. [8, Proof of Prop. 2.2.1). The equality T2P = (P - Q)2 P = P(P - Q)2 = PT2 shows that P commutes with T2, and hence, with T. The proof for Q and R is analogous. The following equality is valid :

TJP = JTP = ( P - Q ) P = (1 - Q ) (P - Q ) = (1 - Q ) JT = T ( l - Q) J .

Since T is injective, we get JP = (1 - Q ) J . The equality JQ = (1 - P ) J can be proved by analogous computations. For R we obtain the sought equation adding the first two,

Lemma 2.8. (i) ( J (x ) , y) = (J(y), z) for any x, y E X. (ii) (J (x) , x ) 2 0 for any x E X.

(J(z), x ) 5 0 for any x E i X .

174 Math. Naohr. 146 (1990)

Proof . There yields PJP = P( l - Q ) J = P(P - Q ) J = PTJa = PT. Thus, ( J ( z ) , z) + (3, J ( z ) ) 2 0 for any z E X since T is positive and P and T commute, (cf. (1) ) . On the other hand

0 = (J(z), ix) + (h, J(z ) )

= 2'- ((a, J(4 - (J(=), 4) for any z E X , (cf. Lemma 2.7.). Consequently, ( J ( z ) , z) = (x, J(;c)), (J(x) , z) 2 0 for any z E X and (J(y) , y) 5 0 for any y E i X . Furthermore, the equality

0 = i * ((iy, JW) + (J(z), iy))

= (J(x), - (9, 44) holds for any x, y E X (or respectively, z, y E iX), (cf. Lemma 2.6.). Thus,

(5) {J(x) , y) = (y, J (x ) ) E Ah for any z, y E X (or, respectively, x, y E i X ) . If we consider now z = x1 + xz, y = y , + y, (zl, y, E X , x,, y, E iX) there yields

= (J(x), y)

= (JW, 2 1 ) + (J(Yz), S) - i - ((J(iyz), 4 + ( J (yJ , ky))

= ( J (y ) , 4,

= (J(Zl) , Y1) + (J(az), Yz) + i- ( (J@l) , iY2) + (J(W, Yl))

(cf. Lemma 2.7. and (6)). Since X + iX is normdense in X the equality (J (x) , y) = (J(y) , z) is satisfied for any z, y E 2%'.

Corollary 2.9. J ( X ) = ( iX ) l , J ( i X ) = XI. This follows from Lemma 2.7. Finally, we can formulate the following

Proposition 2.10. (i) R and (2 - R) are A-linear, injective operakm with the property

(ii) T is injective and A-linear. T = Rllz(2 - R)llS. (iii) J is conjugate-A-linear and injective. Ja =

(iv) T commuta with P, Q, R, J .

For a better geometrical characterization of the operator J we show

Proposition 2.11. The operator J defined above is the unique conjugate-A-linear p r t h l

0 5 R 5 2 , O 5 2 - R 5 2.

For any X , y E 3t? yielas (J(x), y) = (J(Y) , 4.

(v) J P = (1 -&) J , JQ= (1 - P ) J , J R = (2 - R) J .

isometry with the two properties:

( i ) J ( X ) = iX1, J ( i X ) = XI. (ii) ( J ( z ) , a) 2 0 for any z E X , (J(;c), a) 5 0 for any z E i X .

Proof (cf. [8, Prop. 2.3.1). The operator J has the properties (i) and (ii) as we have shown at Lemma 2.7. and Lemma 2.8. Let K be another conjugate-A-linear partid isometry satisfying the conditions (i) and (ii) above. We get (JK) P = J ( l - Q ) R

Frank, One-Parameter Groups 176

= P(JK) , i.e. J K commutes with P. Similarly it can be proved that J K commutes with Q and, hence, with R and T . Consequently,

(6) ( J K ) (RT) = (RT) ( J K ) = ((P 4- &) (P - &) J ) ( J K )

I= (P(1 - &) - Q(l - P)) K

= PKP - QKQ

Since RT is injective, JK = K J . Furthermore, from (6) and (ii) above we draw (RT) ( J K ) 2 0 on X, (cf. Lemma 2.8.). Because of the uniquen- of the polar decom- position and Gince RT 2 0 the equality JK = id% must be wtisfied. Hence, J = K .

Definition 2.12. We take R, X and iX as'at the beginning of this paragraph. There exists an operator S defined on D(S) = X + iX by the formula

S(x + y) := x - y, dc E X , y E ix. The operator S is unbounded, in general. Since iX1 and X I fulfil the conditions "iX1 n XJ = (0)" and "(iX1 + Xl) is dense in X" also there exists an operator F defined on D(F) = i X l + X I by the formula

F(X + y) .-x * - - y, xEiX1, yEX1.

It it3 also unbounded, in general. F and S are c l o d operators.

Proposition 2.13. (i) F = JSJ, F = S*.

Proof (cf. [8, § 6, Prop.]). The equality F = JSJ follows from Corobry 2.10. Taking z E X, y E i X , x E iX1, t E Xl weget

(S(* + y), + t ) = (x - 3, + t ) = 6,x) - (y, t ) = (3s + ?I, F(x + t ) ) .

Therefore, F & S*. On the other hand the equality

(x - y, 2) = ( W + y), 4 = (* + YY S*(Z))

is satisfied for any x E D(S*), x E X, y E iX. If y = 0 there hold@ (2 - S*(x)) E XI. Taking x = 0 we get ( x + S*(x)) E iX1. Conaequently, x = 1/2 - ((2 + S*(x))

176 Math. Naohr. 145 (1990)

+ (z - S*(z))) E D(F). Thus, F = S*. Furthermore,

((2 - R) 8) (a + Y) = (2 - p - &) (a: - Y)

= (1 - P) (a - 9) + (1 - &) (0 - Y) = (1 - Q) (0) - (1 - P) (Y) = (P - &) @ + Y) = JT(x + Y)

for any x E X , y E iX. There follows (2 - R) S = JT on D(S) E %, The other equality is proved similarly. From (i) we have (.IS)* = 8*J = FJ = JS on D(S), (of. (6)). Hence, JS is self-adjoint. On the other hand (RJS) (x ) = T ( a ) for any x E D(S), (of. (ii) above and Lemma 2.7.). Consequently, JS(x) = ((2 - R)lla &-‘la) (2) and x E D(A1la).

Therefore, JS E Alls and since aelf-adjoint operatom are maximal JS = Alla. The second equality can be proved in a similar way.

Corollary 2.14. A114X = A-l /4 ( iXL) , A 1 / 4 ( i X ) = A-114XL.

Remark 2.16. The naturally arising question is why we did not treat the subject of $ 2 in the following way:

First, suppose A being a a-finite W*-algebra. Then, there exists a normal faithful state f on A and X can be completed to a HILBERT space Xf with the inner product f((., .)). Investigating X,, the norm-completion of X in %,, we would get operators P,, Ql, R,, Jl, TI and dl on Xf as done at [8, $5 1, 21. There would “remain” only to restrict all these operators to % s X,,

Secondly, we would generalize these results for non-a-finite W*-algebras A using the constructions of [l, p. 1641.

But, unfortunately, all these operators, in general, do not leave X E Xf invariant and/or are not dl(A)h-linear, at least if A is not commutative (cf. $ 6 of this paper).

Therefore, we can make use of this method only if the existence of the appropriate module operators is already known. So, we will do in the following paragraph.

Moreover, it seems to be possible to generalize the results of this paragraph if A is assumed only to be any commutative C*-algebra. The condition (iii) for X reformulates then as follows :

(iii)’ X is a real self-dual W E R T Ah-submodule of %.

For suggestions in this direction look at $ 4 of the present paper.

§ 3 One-parameter groups and the generalized B.M.S. condition

In this paragraph we consider *weakly continuous unitary one-parameter groups on self-dual HILBEET W*-module# X satisfying a generalized K.M.S. condition with respect to real subspaces X, as they were defined at $ 2 of this paper. Especially, suppotling A to be commutative, we define the p u p {Ait: t E R} in terms of the opera- tor R related to X s %. This group is obtained to be chamcterized by the generalized

Frank, One-Parameter Groups 177

K.M.S. condition. We remark the cl~rle relation of our reaults to the results of M. A. RIEFFEL, A. VAN DAELE [8, $ 31 and to the subject of F. COMBES, H. ZEWL [2, f 31.

Suppose A is a commutative W*-algebra. According to Theorem 2.10., ( l .) , R and (2 - R) are both injective, A-linear, self-adjoint positive operators for which 0 5 R 5 2, 0 5 2 - R 5 2. From [lo, Th. 1.11.3.1 we draw the spectra1 representation of R and (2 - R). The spectral meagure, therefore, is concentrated on the open interval (0,2). Now we can define the one-parameter groups {Pt: t E R} and {(2 - RYf: t E R} since the map I --f ;Iit is correctly defined, bounded and continuous on (0,2) for any t E R. These groups commute and are *weakly continuoud on 35'. Moreover, the equality

(7) is satisfied for any t E R, (cf. Prop. 2.10., (5.)), where the minus sign in the right expo- nent is caused by the conjugate-A-linearity of J .

JRitJ = (2 - R)-it

Definition 3.1. Let Ait := (2 - R)" R-;l, t E R, so that {Ai t : t E R} is a *weakly continuous unitary one-parameter group defined on X.

Proposition 3.2. For any t E R there holds: (i) JAit = AitJ, (ii) Ait(X) = X, Ait(X*) = X I , (iii) Aif[A"4(X)K = [A l /d (X)I ; = [~-1/4(ixl)]; = Ait [A-1/4( iX1)r .

Proof (cf. [8, Proof of Prop. 3.3.1). According to (7) we get JA" = J ( 2 - R)it A-it = 8-itJR-i' = R-it(2 - R)it J

= (2 - R)ifR-itJ = AifJ

for any t E R. Hence, the group {A i t : t E R} commutes with J. Since this group is a function of R it commutes with T and R. Thus, it commutes with P and Q. Consequent- ly, Ait(X) 2 X and also A-"(X) & X since t € R is arbitrarily chosen, i.e., d i t ( X ) = X. The other equality can be proved in the same way. The connection (iii) follows from Corollary 2.14.

Suppose now A is a W*-algebra, noncommutative in general. We remark that an A-valued function g defined on D(g) E C is called to be w*-analytic if g is weak analytic in the sense of [ l , Def. 2.5.21.1.

Definition 3.3 (the generalized K.M.S. condition). Let A be a W*-algebra and X be a selfdual HILBERT A-module. Let X be a norm-complete real subspace of X, X being invariant under the action of X(A),, and satisfying the conditions (i) - (iii) at the beginning of $ 2 . A strongly continuous one-parameter group { 17, : t E R} defined on X is said to satisfy the generalized K.M.S. condition with respect to X if for any x, y E X there exists a map g : C + A defined, bounded and w*-continuous on {z : - 1 5 Im (2) 5 0} and w*-analytic in the interior of this strip, with boundary values given by

g ( t ) = P t ( X ) , Y> B(t - 4 = (Y, Ut(x))

for any t E R.

12 Math. Nachr., Bd. 145

178 Math. Naohr. 146 (1990)

Proposition 3.4. Such a function g as i t is defined by Definition 3.3. is unique. Proof. Suppose there are two functions g and h with the same given properties.

Furthermore, first suppose A being u-finite. Then, there exists a normal faithful state f on A and the complex-valued function f ( (g - h ) (g - h)*) is defined, bounded and continuous on { z : -1 5 Im ( z ) 5 0}, analytic in the interior of this strip with trivial boundary values. By [8, p. 1951 this function must be equal to zero on the given strip, (cf. Remark 2.14.). Consequently, g = h on { z : -1 5 Im ( z ) 5 0).

If A is not cr-finite by [l, p. 1641 there exists an increasing directed net { p a : ar E I} of projections, pa E A, such that paApa is u-finite for any ar E I and w*-limp, = 1,. Investigating the HILBERT paApa-module X u = { p a x , pa(., .) pa} , X , = p,X and the functions pagpa, pahpa, for any ar E I we get pugpa = pahpa on the strip { z : - 1 5 Im (z) 5 0). Hence, g = h on this strip.

Proposition 3.6. Suppose the situation given in Definition 3.3. A *weakly continuous unitary one-parameter group { Ut : t E R} on X satisfies the generaliztd K.M.S. condition with respect to X i f and only i f for any x, y E X there exish a function g: C + A defined, bounded and w*-continum on ( z : -1/2 5 Im (z) 5 0}, w*-analytic in the interior of this strip with boundary values

g(t) = (ot(s), 3) for any t E R, g(t - i /2) E Ah for any t E R.

Proof. If 3, y E X are given we define g(a - bi) := g(a - ( 1 - b) i)* for any num- ber a E R, b E [1/2, 11. This function is defined, bounded and w*-continuous on { z : - 1 5 Im (z ) 5 0} and has boundary values

dt) = (u#(x), 3) J

9(t - i) = (3, Vt(W)

for any t E R. If f ig an arbitrary normal atate on A the construction above give8 US

the SCHWARZ reflection principle applied to the analytic function f(g(z)) along the line Im ( z ) = -I& i.e., f(g(z)) is analytic on { z : -1 5 Im (z) 0) for any normal state f on A. Therefore, g is analytic on this strip.

Conversely, suppose the generalized K.M.S. condition is valid and g is the K.M.S. function. Defining another function h: C +- A by the formula h(z) := g(z - i)* we have two functions g, h with the same properties and boundary values. Since the K.M.S. function is unique (Prop. 3.4.), g = h and g(t - i /2) = h(t - i /2) = g(t - i/2)* for any t E R. Conaequently, g(t - i /2) E A,, for any t E R.

Now we turn on to consider the group {A": t E R} in the cage of A being commutative.

Lemma 3.6. For any complex number z with Im (z) 5 0 we can define the bounded operator Pz. The mup z +- Riz i s a w*-continuous mapping on { z : Im (2) 5 0) and it is w*-analytic in the interior of this set. It is u n i f o d y bounded on horimntal strips of finite width. The same r a u b are true for (2-R)".

Proof. (cf. [8, Proof of Lemma 3.81). The spectral measure of R is supported on the open interval1 (0,2). For any complex number z wit Im(z) 5 0 the function A+ AiZ i s

Frank, One-Parameter Groups 179

bounded and continuous on (0,2). Thus, we can define Riz. Since lAizl 5 2-rm(z) if il f (0,2) and since Im ( z ) 5 0 by supposition there follows that Riz is uniformly bounded on horizontal strips of finite width. Now let a, y c 6%' and {E,: e > 0) be the spectral resolution of R. Then, the restriction of R to (1 - E,) 6%' has its spectrum in ( E , 2 ) and so has bounded logarithm. Consequently, the function (RiZ(l - E,) x , y) is w*-analytic in the entire complex plane. Since R i@ injective, ((1 - E,) a, y) con- verges to (a, y) with respect to the w*-topology on A as e goes to zero. Thus, (Riz(l - E,) a, y) w*-converges to (Riz(a), y), in fact uniformly on horizontal strips of finite width, where Im (z) 5 0, since Riz is uniformly bounded there. Therefore, the map z + (Riz(x), y) is w*-continuous on { z : Im (z) 5 0) and w*-analytic in the interior of this set.

Proposition 3.7. The one-parameter group { A i t : t E R} defined in Definition 3.1. eatis-

Proof (cf. [8, Proof of Prop. 3.7.1). Let a, y E X. We can not set g(z) := (AiZ(x), y)

fie8 the generalized K.M.S. condition with reqect to X.

since this is undefined, in general, on { z : -112 5 Im ( z ) 5 0). But we obtain

2~ = ~ P ( z ) = ( R + T J ) (z) = (R'12(R'~2 + (2 - R)@ J ) ) (a),

and therefore,

(8)

Then

a = R1/2(z) where z = 1/2(R1/8 + (2 - R)l12 J ) (a).

A"(a) = A"Rl/S(z) = (2 - R)'t R1/2-it x (9) 0 and we can define a function

g(z) := ( (2 - A)'* R42-il ( ),Y)

on { z : -112 2 Im ( z ) 5 0) by Lemma 3.6. This function is defined, bounded and w*- continuous on { z : -112 5 Im ( z ) 5 0) and w*-analytic in the interior of this set , 0' ince the multiplication of operators is strongly continuous on bounded sets and since Lemma 3.6. is valid. Using (8) we get

g ( t ) = (d"(x), y) for any t E R.

There remains to show g(t - i /2 ) E Ah for any t E R. The equality

g(t - i l2 ) = ( ( ( 2 - R)" (2 - R)112 R)jt ( x ) , y), cf. (0),

= ( 4 2 - R)'/2 (z), y)

= ((2 - R)'lz ( x ) , d-"(y))

is tlatisfied where

2(2 - R)'le (2) = ( 2 - R)l/' (R112 + (2 - R)'l'J) (a), cf. (B),

= (T + JR) (a?), cf. Prop. 2.10., (6.) ,

= J(TJ + R ) (a?) = 2JP(z) = W(a) .

12*

180 Math. Nachr. 146 (1900)

Consequently, g(t - i /2) = (J(z ) , A-',(y)) E Ah forany t E R because of Corollary 2.9. and Proposition 3.2., (ii).

Proposition 3.8. The group {Ai1: t E R} tk t k unique *weakly continwvus one-parameter group of unitary operdors on X mapping X onto X and satisfying the generalid K.M.S. condition with respect to X .

For this end we show the more general

Theorem 3.9. Let A be any W*-algebra and let 3, X a~ in paragraph two. Let { Vt : t E R} be a *weakly continuow, unitary one-parameter group on X, mapping X onto X and satisfying the generalized K.M.S. condition with respect to X . Then, this q r q is the unique *weakly continuous unitary one-parameter group with these properties.

Proof. Let { Ut : t E R} be another such group on X. We want to show UL = Vt for any t E R.

First, suppose A to be a-finite and let us denote a normal faithful state on A by f . We extend the both one-parameter groups from 3' to the HILBERT space X,, Xf being the completion of X with respect to the norm f ( ( . , -))I/*, (cf. [7, Prop. 2.6.1). The extend- ed one-parameter groups are *weakly continuous and unitary on 2,. They map X, onto X,, where Xf denotes the norm-completion of X in X,. Also they satisfy the K.M.S. condition with respect to X,, what can be seen applying f to the generalized K.M.S. condition for {U, : t E R} and { V , : t E R}, respectively. Consequently, from [8, Th. 3.8., Th. 3.9.1 we derive Vt = Ut , t E R, on X, and by construction also on X.

Secondly, if now A is supposed not to be a-finite, there exists a directed increasing net {pa : a E I} of projections of A satisfying w*-limp. = IA such that paApa is a a- finite W*-algebra for each m € I. Looking for the HILBERT paApa-module X u := { p a x , pa(*, . )pa} , X. := p X . and the groups {paUt : t E R}, {pa Vt : t E R} we get p.Ut = p.Vt on p a x for each m E I, for any t E R. Therefore, Ut = V, for any t E R on X. So we are done.

We remark that for Proposition 3.8. an appropriate generalization of the proof a t [8, Th. 3.8.1 can be realized. But it will be larger than the present one.

Theorem 3.10. Let A be any W*-algebra and let X, X a8 in paragraph two. Let {U,: t E R} be a *weakly continuous one-parameter group of unitaries on X. Suppose that X, is a real submanifold of X being invariant under the action of 2(A) , and such that { U, : t E R} satisfies the generalized K.M.S. condition for X,. Then { Ut : t E R) also satisfies the generalized K.M.S. condition with respect to the smallest real norm-closed sub- space, X , invariant under { Ut : t E R}, containing X , and satisfying X = [Xr. Further- more, X n i X = {O} , so that i f we let X, denote the norm-closure of X + i X , then we can define {AiL: t E R} on Xl using X . Then Xl is invariant under { Ut : t E R} and Ut = A" for any t E Ron 2,.

Proof (cf. [8, Th. 3.9.1). The proof is exactly the same as for [8, Th. 3.9.1. The only difficult point is to prove that X, = [ X , r . Indeed, let y E X,, let (2. : 3ca E X1, a E I} be a bounded net and let ga be the generalized K.M.S. function defined on (2: -1 5 Im (2) 5 0) for the pair (z., 9). Assume that {a. : a E I ) converges to 3c E X with respect to the tl-topology. Furthermore, first assume A to be a-finite, and let f be

Frank, One-Parameter Group 181

an arbitrary normal faithful state on A. Then, applying the maximum modulus prin- ciple on the strip { z : -1 5 Im (2) 5 0} to the function f(ga - gp) we obtain

for any a, /l E I and, hence, { f (ga): a E I} is an uniform CAUCHY net of complex-valued functions being bounded and continuous on the given strip and analytic on its interior. Moreover,

so that this CAUCHY net converges uniformly to a function fo being defined, bounded and continuous on the given strip and analytic on its interior. Moreover, there hold8 fp(t - i)* = fJ t ) = f((U,(a), y)). Since f was arbitrarily cho8en, by [ l , Prop. 2.6.21.1 there has to exist an A-valued function g which is defined, bounded and w*-continuous on ( z : -1 5 Im ( z ) 5 0) and w*-analytic on the interior of this strip, and which satisfies the boundary conditions

g(t - i)* = g(t) = ( U t ( z ) , y} for any t E R. Therefore, if A is a-finite there holds X1 = [X&.

Suppose now A to be non-a-finite. Then, there exists a net { p a : pa E A, a E I) of projections such that paka is @-finite for any LY E I and w*-lim pa = I*. Consequently, pax1 = pa[X,]y for any a E I and, hence, X, = [X&. The desired statement is proved.

8 4 An interpretation on locally trivial Hilbert bundles

In two papers of R. G. Swm [ll] and J. DIXMIER, A. DOUDY [3] it was stated that for every locally trivial &BERT bundle E = (E, X, p , H) with compact basis X the set of all continuous sections r(E) is a HILBERT C(X)-module. Later A. 0. TAEAHASHI [ 121 has proved that for every compact space X the category of locally trivial HILBERT bundles E = ( E , X , p , H) is equivalent to the category of HILBERT C(X)-moduli X . The equivalence is realized by the map X t-f r(t), (cf. [6, Th. 8.11.1). Thus, a C(X)- linear operator on X can be interpreted as an operator on r(t) leaving the fibres in- variant.

Now let X be hyperstonian. Let E be a locally trivial HILBERT bundle over X with the fibre H. For our purpose the HILBERT bundle E must be “full” in that sense that X = r([) has to be self-dual, (cf. Theorem 1.2.). Let X 5 X = r(t) be as in 8 2 of the present paper. If we fix some x E X and regard X , = { ~ ( x ) : z E X } E H , X, = { z ( x ) : z E X) E El, the classical TOMITA-TAKESAKI theorie on El obtains the bound- ed operators P,, &,, R,, T,, J,, A$ (t € R) for the pair (Xz, X,). All these operators can be continued to bounded operators on H. For the global operators defined by (X, X) we get the following property:

(10) ( U ( z ) ) (x) = Uz(3s(x)) for any z E X , (z’ E x = r ( ~ ) , for any U := P, Q, R, T, J , A” (t E R) .

182 Math. Nachr. 146 (1990)

Therefore, in our setting we could define these global bounded operators for the pair (a, X ) by the formula (10) only splitting the appropriate bounded operators for the pairs (Xz, X J , z E X, on X.

The natural question arising from this observation is whether or not this could be done if we merely suppose that X is any compact space, X = r(E) is the set of con- tinuous sections of any locally trivial HILBIORT bundle E over X and X is a HILBERT C(X)h-submodule of X , satisfying the conditions (i) and (ii) at the beginning of $21

If both X and X are self-dual the global operators P, Q, R, T , J can be defined in this way, (lo), by the appropriate local operators. For the strongly continuous unitary one-parameter groups { A t : x E X , t E R} the splitted group {A”: t E R} maps 3f = T(E) into a larger clam of tlectiontl of the HILBERT bundle E , in general. Therefore, this ques- tion must remain to be unanswered.

§ 5.

Let A be a noncommutative W*-algebra and X be a self-dual HILBERT A-module. Let X be as defined in $ 2 of the present paper. The following example shows the difficulties arising if we try to repeat our conception for the noncommutative case.

Example 6.1. Let A = End, ( la) where l2 is the countably enerated Hilbert space.

E A} E 3f is a sequence of finite

Remarks on the noncommutative case

Let X = Zz(A)’ and X = &(Ah)‘, (cf. [a]). We want to defin % the a(A)h-linearprojec- tions P: X -t X , Q : X +iX. If {ak: k = 1, ..., n, length and if Uk 2 U l k + (Uik , U2k E Ah), W e would define

P: { a k : k E N} 3 (air: k E N}, Q : ( a k : k E N } - t { ~ Z c i : k E N } .

by analogy to the (unique) decompotlition of each element a k E A. Now we take the element b := {bk: bk E A, k E N} E ,$(A)’ where

b k : el 4 e k ,

e j - t o , i * l ,

e j - to , j + k ,

b : : e k - + e , ,

(if {ek: k E N} denotes the standard orthonormal basis of Z2(A)). Therefore

P: { b k : k E N} 3 {1/2 - (bk f b:) : k E N} by definition. But the sequence (1/2 - (bk + b:): k E N} does not belong to Z2(A)’. TO see this we look at

1/4 - [ 5 (bk + b:)8] (el) = ( N + 3)/4 - el, N E N. i-1

Obviously the sequence above diverges and, hence, P(b) does not belong to &(A)’. The condition being equivalent to P(a) E Z2(A)‘ for ti certain a E Zz(A)’ is

Frank, One-Parameter Groups 183

The set D(P) of all such elements a E Zt(A)' yields [D(P)];- = X, but D(P) is not nom- dense in X.

If we sum up our results we obtain that P and Q are unbounded A'(A)h-linear opera- tors on X with not norm-dense, different ranges of definition D(P), D(Q), for which X = [ D ( P ) r = [D(Q)]; = [D(P) n D ( Q ) r . The operator R = P + Q is the identity operator on X being A-linear. The operator J is unbounded conjugate-A-linear and it is defined on D(P) n D(Q). J acts there by the formula

J : u = { Q : k E N} + {a:: k E N} . That is, if A is an infinite-dimensional noncommutative W*-algebra we can not repeat the bounded operator approach of this paper! The main reason is the unequivslence of the convergence of the series

m W

zaiut and z u t a i i=l i-1

for sequences {ai: ai E A} E Z2(A) in A, in general.

Now we discuss the case of A being noncommutative and finite-dimensional. In this case A can be awumed to be a complex matrix algebra of finite dimension. The main phenomena will be shown by the following example.

EX8mple 6.2. Let A be a finite-dimensional C*-algebra and let X = A with the A- valued inner product (a, b) = ab*, a, b E X. Let X = Ah. Since A is finitedimensional the norm induced on A = X by the complex-valued inner product /((a, -)) is equivalent to the given =BERT norm on X for any faithful positive (normal) state f on A. By [a, Th. 21 for the positive faithful state f , there exists a bounded linear operator C on X such that:

(i) f((a, b)) = t r ((a, C(b))) for any a, b E 35'. (ii) 0 5 C = C* respectively to t r ((s, .)) and to f((., s}). (iii) There exists C-1 being bounded and linear.

By the equality

t r ((a, c ( 1 A ) b)) = t r ((ab, C(1A))) = f((ab, 1A))

= /((a, b)) = tr ((a, C(b)))

for any a E X, b X there turns out that

C(b) = c ( 1 A ) b for any b E X (and by complex linearity for any b E a). Therefore,

/((a, b)) = t r ((a, b) . C(1A)) for any a, b E X and C(2A) is a self-adjoint positive element of X = A, the eigenvalues of which are all greater than zero.

184 Math. Nachr. 146 (1990)

Since we can not regard X as a real (elf-dual) HILBERT submodule of X and gince the IEILBERT norm is equivalent to the norm f((., .))l/* for any faithful state f on A, we would like to define the projection P: 3if + X as the projection Pf mapping the real HILBERT space {X, Re I((., a))} onto the real ~ E R T subspace {X, Re f((-, .))). But, unfortunately, such a projection PI depend0 on the choice of f ! If f = t r we get Ph: a E X +- 1/2 - (a + a*) E X. This projection is 2(A)h-linear and bounded by one. If f is now an arbitrary faithful pogitive state on A and if we suppoge Pf = Prr by (i) above we obtain the equality (if a = a, + azi E 3e, a,, az E Ah)

for any a E X , b E X, (cf. Proposition 2.4., (1)). Therefore,

0 = tr ((a& - baz) ' Cf(1A))

= tr (b * c f ( 1 A ) ' a,:! - a 2 ' CI(1A) * b)

for any a:!, b E Ah. This is true if and only if c f ( 1 A ) if!l a diagonal matrix. But, since f is arbitrarily chosen, the matrix c f ( 1 A ) can be any self-adjoint positive matrix with strictly positive eigenvalues. This is a contradiction to our supposition Pf = Ptr,

Therefore, P f = Pt, if and only if C,(ZA) is a self-adjoint positive diagonal matrix with strictly positive elements.

Summing up we have to agk whether for each finite C*-algebra A and for any given X and X there exists a faithful positive g t a t e f on A such that the induced projection P f : X + X is 2Z(A)h-linear and bounded by one with reapect to the HILBERT norm on X. Unfortunately, we are not able to answer t h i g question at present.

Ref erenee8

[l] 0. BRATTELI, D. W. ROBINSON, Operator algebras and quantum statistical mechanics, I, Texts and Monographs in Phyaics, Springer-Verlag, New York-Heidelberg- Berlin 1979

[2] F. COMBES, H. ZETTL, Order structures, traces and weights on Morita equivalent C*-alge- bras, Math. Ann. 266 (1983) no. 1, 67-81

[3] J. DIXMIER, A. DOUADY, Champs continus d'espace hilbertiens et de C*-algbbres, Bull. SOC. Math. France 91 (1963) 227-283

[4] M. FRANK, Self-duality and C*-reflexivity of Hilbert C*-modules, ZAA 9 (1990), (2) [6] M. FRANK, Von Neumann representations on self-dual Hilbert W*-moduli, preprint, JINR

[S] K. H. HOFMANN, Representation of algebras by continuous sections, Bull. Amer. Math. SOC.

[7] W. L. PASOEKE, Inner product modules over B*-algebras, Trans. Amer. Math. SOC. 183

[8] M. A. RIEFFEL, A. VAN DAELE, A bounded operator approach to TOMITA-TAKESAIU theory,

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A. 0. TAKAHASHI, Fields of Hilbert modules, Dissertation, Tulane University, New Orleans, La. 1971

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Sektion Mathenzatik Karl-Marx- Universitiit Karl- Narx-Platz DDR - L e i p i g 7010