Math. Nachr. 146 (1990) 187-199

Von Neumann Representations on Self-Dual Hilbert W*-Modnli

By MICHAEL PRANK of Leipzig

(Received March 9, 1987)

Abstract. This paper continues the investigations of [a]. VON NEUMANN algebras M of bounded operators on self-dual HILBERT W*-moduli X possessing a cyclic-separating element z in X are considered. The close relation of them to such special real subspaces X of X as treated in [4] is shown. Under the supposition that the underlying W*-algebra is commutative, a T O ~ A - T A K E - SAKI type theorem is stated for VON NEUMANN algebras on self-dual HILBEET W*-moduli. The natural cone in X arising from the pair (M, 3 ~ ) is investigated and its properties are obtained.

8 1 Introduction

A (left) P r e - b E R T A-module over a certain C*-algebra A is an A-module X equipped with an A-valued inner product, i.e., an A-valued nondegenerate sesquilinear mapping (., a ) : X x X -+ A, (., .) being A-linear at the first argument. The p r e - h E R T A- module X is BERT if it is complete with respect to the norm 11.11 = ll(., - ) I l l 2 . We suppo8e always that the linear structures on A and on X are compatible. For further basic facts concerning HILBERT C*-moduli we refer to [el. A HILBERT A-module X over a C*-algebra A is called selfdual if every bounded module map r : X --f A of the form (-, a) for some a E X. In thi8 paper we restrict our attention mainly to HIL- BERT W*-moduli. For them some more fact8 are known. We need the following ones :

Definition 1.1 [3, Def. 71. Let A be a W*-algebra, X be a pre-H.n.aEBT A-module and P be the set of all normal states on A. The topology induced on X by the semi- norms

f K . 9 .))r’a ,

f ( ( . ,a ) ) , f € p, a E X,

f E p 9

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

i8 denoted by t2.

Throughout thi8 paper we use the following notation. If X is a subset of the HLL- BERT W*-module X, [XI;- denotes the set { A - x: 1 E R,, o E X,) where X , i8 the t,-completion of the set {x E X: llall 5 I}.

188 Math. Nechr. 146 (1990)

Theorem 1.2 [3, Th. 91. Let A be a W*-algebra and X be a H~LBERT A-module. The

(i) d is 8elfdual. (ii) The unit ball of X is complete with respect to the topology zl, i.e., X = [Xr. (iii) The unit ball of d is complete urith respect to the topology z2.

Corollary 1.3 [3, Cor. 111. If A is a W*-algebra and d is a self-dual HILBERT A- module the linear span of the range of the A-valued inner product on X becomes both a W*-subalgebra and an ideal in A.

Theorem 1.4 [6, Prop. 3.10.1. Let A be a W*-algebra and X be a self-dual HILBERT A-module. Then, the set EndA (a) of all bounded A-linear operators on X is a W*-algebra.

These facts make clear that in the case of d being a selfdual HILBERT W*-module the spectral theorem ([Q, Th. 1.11.31, [6, Th. 7, 8,9]) is valid for each self-adjoint ele- ment of EndA (d). Moreover, there exists a polar decomposition for each element of EndA (a) in End* (a). This is of importance for the existence of certain operators arising from some special real subspaces X of self-dual HILBERT W*-moduli X, as they were treated in [4].

We remark that for a given W*-algebra A any considered selfdual HILBERT A- module X a n be aasumed to have A as the linear span of the range of its A-valued inner product, (cf. Corollary 1.3). Otherwise we would change A to this linear span of the range, B c A, and we would consider X as a HILBERT B-module.

Let A be a W*-algebra and d be a selfdual HTLBERT A-module. 5 2 of this paper investigates (generalized) VON NEUMANN algebras M on 3C po$$eMing a cyclic-separating element x in X. The definition of a cyclic element for M is modified in a nonobvious way. We show the relation between VON NETJMA” algebras on d possessing a cyclic- separating element and such special real subspaces X of X as they were investigated in [4]. In addition, a generalization of KAPLANSKY’S density theorem for *-algebras of bounded operators on d is stated. Under the supposition that A is commutative, we consider the partial conjugate-A-linear involution J and the strongly continuous unitary one-parameter group {A”: t E R} arising from that real subspace of d which corresponds to the pair ( M , x ) in $3. They would fulfil the conditions of TOBXITA- TAKESAEI’S theorem for M and M’. $ 4 deal8 with the natural cone b d connected with the pair (M, 2). We investigate the main properties of this cone.

following conditions are equivalent:

§ 2

Let A be an arbitrary W*-algebra. We would like to show that VON NEVB~A” algebras of bounded A-linear operators on a selfdual &BERT A-module d, which po-6 a cyclic-separating element x E d, are closely related to certain qecial real subspaces X of d, as they were treated in [4]. The key point of this paragraph is Definition 2.6 modifying the notion of a cyclic element for a VON NEUMANN algebra over a self-dual HILBERT W*-module. Thus, new aspects are produced in the discussions. At the end of this paragraph we state a generalization of KAPLANSKY’S density theorem for *- algebras of bounded A-linear operators on self-dual HILBERT W*-moduli over A.

Von Neumann algebras on self-dual Hilbert W*-moduli

Frank, Von Neumann Representations 189

Definition 2.1. Let A be a W*-algebra and be a self-dual HILBERT A-module. A C*-subalgebra M of EndA (a), (the set of all bounded operatort3 on X being A- linear), coinciding with its bicommutant M” is called a “generalized” VON NEUMANN algebra.

Corollary 2.2. Let A be a W*-algebra and X be a self-dual HILBERT A-module. Any “generalized” VON NEUMA” algebra on is a W*-algebra, i.e., the wurd “generalized” is superfluous in Definition 2.1.

Proof. Since EndA (a) is a W*-algebra [6, Prop. 3.101 there exists a normal faithful representation n of EndA (X), i.e., n(EndA (a)) = n(EndA (a))”. Moreover, n(M)“ E - n(EndA (2))” = n(EndA (a)) since the relation n(EndA (a))‘ n(M)’ holds for the commutants. By supposition M = M”, and hence, n(M) = n(M)”. Consequently, the repreflentation n is faithful and normal for I, too.

We state now an important topological property of these VON NETJNANN algebras.

Definition 2.3 [3, Def. 161. Let A be a W*-algebra and 3f be a &BERT A-module. The bounded net {B,: B. E EndA (,X), d~ E I} converges to B E EndA (X) relative to the topology X 3 if there exists

w*-lim (B,(a), b) = (B(a), b) .€I

for any a, b E X. (For other topologies of. [3]).

Proposition 2.4. Let A be a W*-algebra and X be a self-dual HILBERT A-module. Let M be a C*-subalgebra of EndA (X). Then M is a VON NETJMANN algebra if and only if the unit ball of M is complete with respect to the topology X3.

Proof. We take a bounded net with the following properties: (i) {B,: B, E M, (ii) BnC = CB, for any C E M’, a E I .

E I) converges to B E EndA (X) relative to the topology x8.

Therefore, 0 = w*-lim ((BUG - CB.) ( x ) , y)

, € I

= ( ( B e - CB) (a), y)

for any x , y E X. Thug, B E I” = M. On the other hand, suppose that the unit ball of M is complete with rapect to the

topology X,. We consider a faithful normal representation z of EndA (X), (cf. Theo- rem 1.4). It is also a faihhful repregentation for M. By [6, Remark 3.91 the topology X, coincideg with the weak topology on EndA (a). Therefore, the unit ball of n(M) is complete with respect to the weak* topology, and hence, n(M) = n(M)” by [l, Th. 2.4.111. Consequently, M = M“ since iz is a normal representation of EndA (X).

By [6, Remark 3.91, [3, Th. 91 we conclude that the topology X 3 coincides with the weak* topology on the unit ball of M.

190 Math. Naohr. 146 (1990)

In further discussions we are especially interested in VON NEWN algebras M on selfdual HILBERT W*-moduli X posseaging somewhat like a cyclic-separating element x E X similarly as a-finite VON NETJMA” algebras possess one in certain HILBERT spaces.

Definition 2.6. Let A be a W*-algebra, X be a self-dual HILBERT A-module and M be a VON NEWN algebra on X. An element x E X is called to be cyclic for M iff [Mx]; = X. It is called to be separating for M iff B ( x ) = 0 for some B E M implies

Example 2.6. Let A be an arbitrary infinite-dimensional W*-algebra. We denote by Z,(A) the set of all bounded sequence$ of elements of A. The algebra l,(A) can be considered aa an involutive algebra of operators on the self-dual &BERT A-module .&(A)’, the Adual of .&(A) (cf. [3, Th. 22]), if we define for a given element a = {ai : i E N, ai E A} of lm(A):

B = 0.

a: Z2(A)’ --f Z2(A)’, a(b) := a({bi: i E N}) := {biai: i E N}, a* : = {a; : i E N} .

Obvioualy, each element a E Z,(A) is bounded with sup {llaill: i E N} and A-linear, (cf. [6, Prop. 2.31). Thus, ,!,(A) 5 EndA (Z2(A)‘). Moreover, Z,(A) 5 EndA (,$(A)’) is a VON NETJMA” algebra. Its commutant is l,(X(A)), where 2 ( A ) denotes the centre of A. If we consider a sequence of numbers {Ai: i E N, l i + 0, ili E C} E 12, we get an element x := {ili - 1A: i E N} E ,?,(A) c Zg(A)’ being cyclic-separating for Zm(A). Let US

remark, that x is not cyclic for lm(2(A)), the commutant of Z=(A), if A is not com- mutative. Moreover, there does not exist any “norm-cyclic” element for l,(A) in

Proposition 2.7. Let A be a W*-algebra, X be a self-dual HILBERT A-module and M be a VON NEWN algebra on X . If an element 3c E X is cyclic for I, it is separating for M . If an element x E X is separating for M’ and A[Mxl-; 5 [Mxl-;, it is cyclic for M.

Proof. First, we consider an element B‘ E M such that B’(x) = 0. Then, B’(B(2)) = (B‘B) ( x ) = (BB’) (2) = B(B’(x)) = 0 for any B E M. If x is cyclic for M the ele- ment B‘ E M’ must be equal to zero.

Secondly, if x is separating for M’ and A[MxK 5 [Mxl-; there exists a projection P E EndA (a), P’: X +- [MxI;-, (cf. Theorem 1.2). The projection P‘ is contained in M’. Therefore, (id;, - P‘) (2) = 0, and hence, P’ = id;, since x is supposed to be separating for M‘.

We know that a VON NEUMA” algebra M poaesses a cyclic-separating element in a certain -BERT space if and only if M id u-finite, [l, Prop. 2.5.61. In our setting this statement is not in all cases true. However, we can state the fovowing

Lemma 2.8. Let A be a W*-algebra, X be a selfdual HILBERT A-moduZe and M be a VON NEUMANN algebra on X possessing a cyclic-separating element in X . Then:

(i) If A is u-finite, M is u-finite. (ii) If M is u-finite, the centre 2 ( A ) of A is u-finite. (iii) The VON NETA~ANN algebra M is not necessarily a-finite.

1 2 W .

Frank, Von Neumann Representations 191

Proof. If a E 2 is a cyclic-separating element for M and if {E.: E. E M, LX E I} is a set of mutually orthogonal nonzero projections, we obtain

(a, a) 2 Z (Ea(a) , Ep(z)) . . B d

= 2 (Ea(a), Ea(a)) 2 0. a d

Since A is a-finite by assumption, there exists a faithful normal state f on A. There- fore,

+m > /((a, 2)) 2 Z f ( (Ea(a) , Ea(Z) ) > 0) .€I

and +oo > f ( (E , ( z ) , Ea(a) ) ) > 0 for any LY E I. Thus, the set I must be countable and M is a-finite. This proves (i).

To show (ii) we remark that

{ a . id,: a E 2 ( A ) } = 2(EndA (X)) E 2 ( M ) ,

(cf. [7, Cor. 7.10, Prop. 8.11). Consequently, if M is a-finite, 2(A) has to be a-finite, too.

The third statement can be shown considering a non-a-finite W*-algebra A, first, a d a self-dual HILBERT A-module X with A-valued inner product (a, b) := ab*, a, b E A, and secondly, as a VON NETJIKA” algebra M on itaelf (A = X ) with cyclic-separating element lA E A, where the elements of M are defined as multiplications of A with elements of A from the right. So, the proposition is proved.

Now we describe the relation between VON NEUMANN algebras M on self-dual H h - BERT W*-moduli X posseaing a cyclic-separating element x E 3e and those special real subspacea X of X, as they were treated in [4], under the supposition, that A is commutative.

Proposition 2.9. Let A be a commutative W*-algebra, X be a selfdual IEBT A- module and M be a VON NETSMA” algebra on X possessa’ng a cyclic-seprding element x E 3. Let X := [Mfl];. Then X is a real subspace of X being invariant under the action of A and satakfying the conditions:

(i) X n i X = {0}, (ii) X + i X is normdense in 3e.

Moreover, [M&]; 2 iX1. Proof. For any B E Mh, B‘ E Mi, the following two equalities are valid:

@(a), B’(a)) = (B’(a), B(a)) , 0 = (iB(a), B’(a)) + (B’(z), iB(x)) .

Therefore, B(x) E iX1 for any B E Mi. Now we obtain the relation [ M i x 6 obvious computations, (cf. [4, Prop. 2.21). From the relation

iX* by

M’a G iX* + X l S ( X n i X ) l

we derive (X n i X ) l = X since it! E X i8 cyclic for M’ by Proposition 2.7 and since [(X n i X ) l K = ( X n i X ) l by [4, Prop. 2.21. Consequently, X n i X = (0). More-

192 Math. Nachr. 145 (1990)

over,

and we get that X + i X h 4 ~ to be norm-dense in d since a E X is cyclic for M. The proposition is proved.

Therefore, if A is a commutative W*-algebra, we can apply the results of [4] to X = [Mp];. We will make use of this in Q 3 of this paper. We remark that the Pro- position 2.9. is also true if A is noncommutative. But the proof is much more difficult in this caae.

At the end of the second paragraph we prove a generalization of KAmANsKY'd density theorem we need later.

Definition 2.10. Let A be a W*-algebra and {a, (., .)} be a HIIBERT A-module. We say that a bounded net {B.: u E I, B. E End; (X)} converges to B E End; (X) relative to the topology a - X ; iff there exists

M a 5 x + ix & 3e

W

w * - h L' ( ( B a ( 4 , &((En)) + (BX(an), BX(=n))) acZ n = l

= E (@((En), B(an)) + (B*(zn), B*(an,>) n = l

8

for any sequence a,: a, E X, n E N, llzn/ls < +-). Theorem 2.11. Let A be a W*-algebra and {X, (., .)} be a self-dual HILBERT A-moclule.

Let N be a self-adjoint algebra of bounded, A-linear operators on d, and let M be the VON NEU.MA" algebra arising as the linear hull of the X8-cmnpletion of the unit ball of N. Then, the unit ball of N i s u - X,1-dense in the unit ball of M.

Proof. First, suppose A to be a-finite. Let f be a normal faithful state on A . Con- sidering the extensions of the operator algebras N and M from EndA (X) to Endc (XI), cf. [6, Th. 2.81, (where X J denotes the closure of with respect to the norm f((.,.) )l's), we can apply KAPLANSKY'S density theorem to N E End, (XI ) . Therefore, the unit ball of N 5 End, ( X J ) is a-strong*-dense in the unit ball of M" Endc (X,), the bicommutant of M in Endc (XJ) . Consequently, the same is true if we change M" to M M". Since the state f is faithful and normal we get the desired statement in the case of A being a-finite.

Secondly, let A be non-a-finite. Then, by [l, p. lSa] there exists a directed increasing net { p a : u E I} of projections of A such that paAp,, is a-finite for any u E Z and that there exists w*-limp. = lA. As it has been shown, the unit ball of p.N is u - x;- dense in the unit ball of p,M for any u E I . Hence, the sought statement can be derived easily since (u E I is arbitrary and w*-limp. = 1 ~ .

{ n = l

s 3 A Tomita-Takesaki type theorem

Throughout this paragraph A is assumed to be a commutative W*-algebra. We want to show that the TOMITA-TAKESAKI theorem is valid for VON NEUDU" algebras M on self-dual HILBERT A-moduli X possessing a' cyclic-separating element (E E X, for

Frank, Von Neumann Representations 193

its commutants and for the derived from the pair ( M , x ) modular operatow J and {Ait: t E R} on X, (cf. Prop. 2.9., [4, Prop. 2.10, Def. 3.11). We remark that this theo- rem was stated in [2, 0 31 for the special case M = EndA (X).

Theorem 3.1. Let A be a commutalive W*-algebra, X be a self-dual HILBERT A-module and M be a VON NE~IMA" algebra on X posses&g a cyclic-separating element x E 3f. If X is defined in terms of the pair (My x ) a8 at Prqo"ition 2.9, and i f J and {A": t E R} are defined by X , ([a, Prop. 2.10, Def. 3.1]), there hold:

(i) JMJ = M', (ii) d"MA-" = M for any t E R.

First, we will prove the theorem under the assumption that A is a-finite. There are two ways to do that. The first one ig to use the constructions from [7, 0 41 looking al- ways for modifications needed in the proofs since, in general, norm-completeness of X it3 changed to tl-completeness of the unit ball of X. This can be done, but we will not do so in this paper. From the proof in [7, 8 41 we give only the generalization of the key Lemma 4.3 a t the end of thig paragraph because it seem of more general interest. The second way of proving Theorem 3.1 bases on the following lemma:

Lemma 3.2. Let A be a commutative a-finite W*-algebra and let X be a self-dual HIL- BERT A-module. Let M & EndA (a) be a VON NEIJMA" algebra p088essing a cychk- separating element x E X. Let f be a faithful normal state on A and denote by XI the norm- closure of X with respect to f ( ( - , .))l12. Then, any bounded, A-linear operator on X can be continued to a (unique) bounded linear operator on XI with the same operator w m , and moreover,

(i) M' & EndA (a) is identical with M' (ii) M = M" in End, (XI ) .

Proof. Because of [6, Th. 2.81 we have only to prove (i) and (ii). For a fixed faithful normal state f on A we consider a projection 2 E Endc ( X I ) commuting with any B E M E EndA (a) (1) ZB(x) = BZ(x) for any B E M.

If x = x1 + x2 is the unique decomposition of x in X I with respect to the subspaces ZXl , x1 E Z X l , and (id - 2) X I we draw from (1)

B(a,) = Z(B(xl) + B(x,)) for any B E M y

Endc ( X I ) ,

End, (;re/). Then, we get

i.e., B(xl) E 2 X l and B(x2) E (id - 2) X I for any B E M. Since Z is the identical operator on (ZXl n X) and since 2 commutes by aeeumption

with any {a - id,: a E A} M, the projection Z E Endc (3fl) is the extension of a (unique) projection 2 E M' E EndA (a). Thus, by [6, Th. 7,8] and [9, Th. 1.11.31 M 5 EndA (X) is identical with M' E End (XI) . The second statement above is now obvious.

We remark that in the case of A being noncommutative the statement (ii) is true, whereas (i) fails to be true.

Lemma 3.3. If A is a-finite the statement of Theorem 3.1 is true.

13 Math. Nachr., Bd. 145

194 Math. Naohr. 146 (1990)

Proof. If f is a faithful normal state on A we consider M G Endc ( X f ) and 3c E 35' G XI, (cf. Lemma 3.2). The appropriate operators Jf and {A:: t E R} are the extensfom of the operators J and (AiL: t E R} from 35' to X f , (cf. [4, Prop. 2.11, 3.81). By "7, Th. 4.21 we get

A;%AT" = M for any t E R on X f for M, M' E Endc ( X f ) . Applying Lemma 3.2 the desired lemma yields.

JfMJf = M ' ,

Now there remains to prove Theorem 3.1 in the case of A being non-a-finite.

Lemma 3.4. I f A is non-a-finite the stutement of Theorem 3.1 Q true, loo.

Proof. Let X be defined in Aermg of (M, a) a$ at Proposition 2.9. The operators J and {A i t : t E R} are defined for X as in [a, $8 2, 31. By [l, p. 1641 there exidts an in- creasing directed net {-pa: u E I } of projections of A such that paApa is a a-finite W*- algebra for any u E I and that w*- lhpa = 1A. Investigating for a fixed u E I the linear spaces Xa = {pa%, pa (-, .) - p a } and X , = { p a x , pa. (., . ) . p a } , we obtain operators J , AiL (t E R) on Xa. The bounded operator J satisfies the condition$ of [4, Cor. 2.91 with respect to p a x . The same conditions are valid for the bounded operator p,J. Consequently, by [4, Prop. 2.111

paJ = Ja on X a for any u E I .

Similarly, the operators paAu and A:' (t E R) both satisfy the generalized K.M.S. condition with respect to X a = pax, and moreover, paAit(paX) = A:;(paX) = p,X for any t E R. Therefore, by [4, Prop. 3.81

padit = A:t on X , for any u E I , any t E R.

X3-complete by Proposition 2.4 and since for any B E M, any y E 35' there exidts Since A:tBA;iL E EndA ( X ) for any B E M, any t E R, since the unit ball of M is

t,-lim A~Bd;"(p,y) = AitBA-iL(y), t E R , aeI

we obtain

3-3-lim AtBA:' = AiLBd-" E M , t E R , a €1

for any B E M . Therefore, LI'LMA-'~ = M , t E R, since t E R can be changed to ( - t ) E R in the relation (2).

Similarly, we can prove the relation JMJ = M' using the X3-completenesa of the unit balb of M and M , respectively. The Theorem 3.1 is proved.

Corollary 3.6. [Mkzr = iX1. Proof. From Theorem 3.1 we draw Mix = JMhJx = JMG. By 14, Cor. 2.9.1 the

relation J ( X ) = iX1 holds. Since [MhxI;- = X by definition and since J is injective, [Mi=]; = iX1.

(2)

In addition we like to state the lemma "7, Lemma 4.31 in a generalized form.

Lemma 3.6. Suppose the situation given at Theorem 3.1. Let B' E M;. Then f m any I E C with Re ( I ) > 0 there ezists a unQue B E Mi, such that

2 (C(Z), B'(z)) = I * (C(Z), B(x)) + X (B(z) , C(Z)) for any C E MA.

Frank, Von Neumenn Representations 195

Proof. First, let A be a-finite. We assume Re ( A ) = 1 and 0 5 B 5 1 rtince the sought equality i0 real linear. Let f be a fixed faithful normal positive state on A exist- ing by [ l , Prop. 2.5.61. Define two normal functionah r and rB on kf by the formula

r(C) = f (2 - (C(=), Wz))) , r e v ) = f(J. * (Q(a), + 3 * (B(@, C(Z))),

for any given B E Mh, for any c E Mh. Because of [6, Remark 3.91, [3, f i r . 171, Pro- position 2.4 and Corollary 2.2, the functionah r and rB are in the self-adjoint part of the predual of M. Let

This set is obviously convex. Since the map B -+ rB is continuous with respect to both the w*-topology on M and the weak topology on the predual of M, the set V is weakly compact.

Assume r 4 V. Then, by the HAHN-BANACH separation theorem there exists a n element D E Mh such that

V := ( rB: B E Mh, llBll 5 1).

(3) TB(D) < r(D) for any B E Mh, llBll 5 1.

Let D = U . ID1 be the polar decomposition of D E Mh existing since SYP is self-dual, (cf. Prop. 1.4). U is self-adjoint and commutes with ID1 since D is self-adjoint. Taking B = U we draw from (3):

1/2 - f ( A * (D(x) , U ( 4 ) + 2 * (W4, D(*))) < f((D(Z), B’W) = f(”B’)”2 (4, (B’P2 (3,)) 5 f((lD1 - (B””8 (a), (B‘11’2

= f(<lDl * B’(a), @) 5 f(Wl * (=), 4) = f ( (D(a ) , g(2))) = 11.2 * f ( A * (D(*), u(e)) + 2 * (o(z), WJ))).

This is a contradiction. Consequently, r E V, i.e., there exists an element B, E I llB,ll 5 1, for which

(4) 2 - f ( ( W ) , B’(4)) = f ( A (C(*), B / W + 1. - (B/(Z)Y C(Z)))

for any C E Mh. We define

D := (2 - (C(Z), B’(z)) - A - (C(Z), Bf(Z)) - A * (Bf(Z), C ( 0 ) ) ) - C .

If C E Mh, the element D belongs to Mh. Putting D to (4) we obtain

(6) f ( 12 (C(z), B’(*)) - A * (C(z), Bj(x)) - 2 - (B/(z) , C(z))I8) = 0

for any C E MI. Since f is faithful and positive we get the desired equality in the case of A being a-finite. By the way we have shown that Bf E Mh does not depend on f.

We remark that the only place is (4) where the a-finiteness of A is needed, because we did not know whether or not Bf depends on the choice of f . Now the restriction on A to be a-finite can be dropped in an obvious way using [l, p. 1641. The lemma is proved.

Let us remark that the assumption A ha0 to be commutative is not used in this prove.

13*

196 Math. Nachr. 146 (1090)

g 4.

Throughout this paragraph A is assumed to be commutative. Let X be a self-dual HILBERT A-module. We consider VON NETJWN algebras M on X possessing a cyclic- separating element x € X. The modular involution, the modular operator and the appropriate modular group of automorphisms associated with the pair (M, x) are denoted by J, A and {Aif: t E R), respectively. For details see [4] and Proposition 2.9. The aim of thid paragraph is to show the geometrical properties of the natural positive cone 9 E X arising from the pair (M, x) in some sense similarly as in the case A = C,

Natural positive cones in self-dual Hilbert W*-moduli

cf. [l].

Definition 4.1 (cf. [l, Def. 2.6.251). For a pair (M, a) we consider the set 90, the z,-completion of the set {BJBJ(x): B E M, IIBJBJ(x)ll 5 1). Then, the natural positive cone associated with the pair (M, x) is defined as 9 := [P0]T.

Proposition 4.2. The closed set 9

(i) 9 = [A1/4Mtx], = [A114[M+xr],, X has the following properties:

= [ ~ l - l / ~ M ; x r = [A-1/4[M;~]T],,

where M+ and M; are the self-adjoint positive par& of M and M', respectively. Therefore, 9 is a convex cone.

(ii) A i t 9 = 9 for any t E R. (iii) For any y E b the equality J(y) = y holds. (iv) If B E M we get BJBJ(9) 2 3'. Proof (cf. [I , Prop. 2.6.261). Let Mo 2 M be the *-algebra of the entire analytic

elements of the group {A": t E R). For any B E Mo there holds

(6) A 1/4BB*(x) = A 1/4BA -1/4A 1/4B*d -1 /4 (3)

= A 1/4BA-1/4(A - 1/4BA I/'¶)* (a)

- - A1/4BA-f/4 JAl/a(A-l/4BAl/4) ( 2 )

= (A'/4BA-1/4) J(A'/4BA-l/4) J @ ) .

Since Al/4M,A-1/4 = M, and because of Theorem 2.11, the relation (6) yields

9 5 [d1l4M+zl; S [L~~/~[M+z]T];.

On the other hand, [&=I;- = [M+x]T by Theorem 2.11. For an arbitrary y E [M+xK we take such a bounded net {B,,(x): oc E I , B, E Mo+) that tl-lim BU@) = y. Then, the equality (6) implies dl/4BU(a) E 9. But

tl-lim JA1/aBU(z) = z,-lim &(as) = y = Jdl/a(y) . ,€I a d

Therefore,

("(y - B,(x)), A1/4(y - B,(z))) = (J(y - B,(x)) , JA'\Z(y - Ba(z)))

and A1/4y E 9, i.e., [~ll/~[M,ae]T]; G 9. Finally,

9 = [dl/'M+~:l;- = [A1/4[M+~r],.

Frank, Von Neumann Representations 197

If 9’ denotes the natural cone associated with the pair (M’, a!), there holds

B‘JB’J(z) = J(JB‘J) J ( J B J ) (a) = (JB’J) J(JB‘J) J(a)

for any B E M’ since J B J E M, i.e., b’ = 9. Since A-1 is the modular operator with respect to the pair (M’, a!), we obtain

9 = 9‘ = [&1/4M‘ + z 11 - = [A-l/4[M;a!];-];.

The first gtatement is proved.

To prove the second one keep in mind that &tA1/4M+z = dl/4diN+z,z = A1/4A”M+d-ita! = A1/4&3.

If B, C E M the third statement follows from the equality

J(BJBJ(a!)) = (JBJ) BJ(a!) = B(JBJ) J(a) = BJBJ(z),

whereas (iv) can be derived from

(BJBJ) (CJCJ) (a!) = BC(JBJ) (JCJ) (a!) = BCJ(BC) J ( z ) .

So, the proposition is proved.

Proposition 4.3. The following relalions are valg:

(7) [ M + z r = {y E 2: (9, x ) 2- 0 for any x E [M$I;-} , [ M i z K = {y E x: (Y, 4 h 0 far any x E [M+a~lr}.

Proof. First, suppose A to be a-finite and let f be any normal faithful state on A. We denote by [M+a!L the closure of [M+a]; with respect to the norm /((a, .))llr on X . The set [M;a!I, is defined analogously. Then, by [l, Prop. 2.5.271 the following set identities are valid on the HILBERT space X f , the closure of X with respect to the norm f((., .))1/2:

[M+a!L = {y E 2,: f((y, x ) ) I 0 for any x E [M>aI,} , [M;zlf = {y E 3 f f : f((y, x ) ) 2 0 for any x E [M+a!lf}.

Since f is an arbitrary normal faithful state, we obtain

[M+a!K = {y E 3 f : f((y, x ) ) h 0 for any x E [MLar , any f E P} , [Mia!]; = {y E 2: f ( (y , 2)) 2 0 for any x E [M+aK, any f E P} .

Thus, we get the sought relations (7). Secondly, let A be non-a-finite now. There exists a directed increasing net {pn : a E I}

of projections of A with the properties that pa Apa is a a-finite W*-algebra for any a E I and that w*-lim pn = 1A. From the first part of the present prove one derives

pa*[M+ZK = {y E pax: (y, x ) h 0 for any E pa*[M>z];-;-) , pa.[M;a!K = {y E p a x : (y, z) 2 0 for any x E pa.[M+Zr}.

for any (x E I, if paM on pax is considered. Since a E I is arbitrarily chosen we get the desired relations (7).

198 Math. Nachr. 146 (1990)

Proposition 4.4. (i) The cme 9 2;9 df-adjoint, i,e.,

= bv :=

(ii) b n (-3') = {O}. (iii) If J(Y) = y for a certain y E X there exzkt-s a unique deccnnposition y = y1 - y2

(iv) The linear hull of b i8 X . Proof (cf. [l, Prop. 2.5.281). (i) First, suppose A to be a-finite and let f be an arbi-

trary faithful normal state on A. If we consider the HILBERT space X, and the cone $1 G 31, (the closure of 3' and X, respectively, with reapect to the norm /((., .))1/2),

there turns out by [l, 2.3.281 that

X : ( y , x ) 2 0 foranyx E b}.

with y1,2/0 E 9, and y1 I y2.

3'1 = 8'; = {y E X I : f((y, x ) ) => 0 for any x E bf} . Since f is arbitrarily choaen, we obtain

9 = {y E X: f ( ( y , x ) ) h 0 for any x E 9, any f E P} = {y E X:(y,z) 1 Oforanyx E 3') = bv

as desired. Secondly, if A is non-a-finite there exists a directed increclsing net {pa : u E I} such that paApa is a-finite for any u E I and that w*-limpm = lA. Considering p a d & p,X we get pap = papv for any oc E I. That implies b = bv.

(ii) If y E 3' n ( -2) we draw from (i) that (9, y) = 0, and hence, y = 0. (iii) Assume J(y) = y. The cone b is a closed convex subset in the self-dual HIL-

BERT A-module X having the property that the subset Po = {y E 8: llyll 5 1) is tl- complete. Therefore, there exists a unique element y, E b such that

(8)

Let US denote y2 := y - yl. For any x E b, A > 0 the inequality

(y - y1, y - 91) = inf {Q - 2, y - 2): x E b}.

(Y - 91, Y - 91) 5 (Y - (Yl + A * 4, Y - (91 + 1 2))

0 s 1 ((Y,, 2) + (x ,y2)) + 28 - (x , 2).

holds since (yl + ;(x) E b. Therefore,

Since A > 0 is arbitrarily chosen the inequality (yp, 2) + (x , yz) 2 0 hm to be valid. But, since E b there hold8 J(y,) = y2, J(x) = x , and hence, (yp, x ) = (J(y2), J(x)) = (y,, x)*. Thus, we get y2 E 3' since x E b was arbitrarily chosen and (i) ie valid. Consequently, y = y1 - y2 where yl, ya E 3'.

Now let us show that y, 1 y,. Since (1 - A) y, E b for any 0 5 1 5 1 we get

(Y2, Yz) I (Yo - A * YlY 3 2 - A * Y1)

because of (8). This is equivalent to

A2 * Ql, 91) - 1 ((Yl, 92) + (y*, Yl)) r 0. Finally, (y,, y,) 5 0, and since yl, y2 E b = bv, we obtain (yl, y2) = 0.

Frank, Von Neumann Representations 199

To show the uniqueness of such a decomposition we assume the existence of two decompositions

Y = Y 1 - Y 2 , Yl ,Y,EcfP, Y l l Y Z ,

y = x 1 -%,, x l , x , E 9 , % l I % .

(Yl - x1, Y1 - XI) = (Yl - x1, Yz - 4 = -(% yz) - (Yl, x,> 5 0.

Then y1 - x1 = y, - x,, and furthermore,

Therefore, y1 = xl, y, = x, and such a decomposition is unique. (iv) If an element y E 3t is orthogonal to the linear hull of 9, there must be (y, y) = 0,

and hence, y = 0. There only remains to remrk that the unit ball of the linear hull of 9 is z,-complete by construct,ion. The propoctition is proved.

Corollary 4.6. (i) I f y E 9, y is cyclic for M if and only i f y is separating for M. (ii) If y E 9 i8 cyclic (and hence, separating) for I, for the modular involution J y and

for the natural positive cone PU U880CZded with the pair (M, y) the relations J = J,,, 9 = 9u hold.

The prove of this corollary is analogous to that of [l, Prop. 2.3.301, and thus, will be omitted.

Ref erenee8

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

[2] F. COMBES, H. ZETTL, Order structures, traces and weights on Morita-equivalent C*-algebras, Math. AM. 366 (1983) no. 1, 67-81

D] M. FRANK, Self-duality and (?*-reflexivity of Hilbert C*-modules, ZAA 8 (1990), (2) [4] M. FRANK, One-parameter group arising from some real subapaces of self-dual Hilbert W*-

moduli, preprint, JINR Dubna (U.S.S.R.), 1987, (E6-87-96), Math. Nachr. 146 (1990) [6] R. M. LOYNES, Linearoperators in VH-spaces, Trans. Amer. Math. SOC. v. 166 (1966) 167-180 [6] W. L. PASCFIKE, Inner product modules over B*-algebras. Trans. h e r . Math. SOC. v. 183

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

[S] M. A. RIEBBEL, MOWA equivalence for C*-algebras and W*-algebras, J. Pure and Appl.

[9] S. SAKAI, C*-algebras and W*-algebras, Springer-Verlag, Berlin- Heidelberg-New York

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