Math. Nachr. 175 (1995) 61 -83

Hilbert C*-Modules over Monotone Complete C*-Algebras

By MICHAEL FRANK of Leipzig

(Received September 10, 1992) (Revised Version September I , 1994)

Abstract. The aim of the present paper is to describe self-duality and C*-reflexivity of Hilbert A-modules dl over monotone complete C*-algebras A by the completeness of the unit ball of & with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results of H. WIDOM [Duke Math. J. 23,309-324, MR 17 # 12281 and W. L. PASCHKE [Trans. Amer. Mat. SOC. 182, 443-468, MR 50 # 8087, Canadian J. Math. 26, 1272-1280, MR 57 # 104331. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. M. OZAWA [J. Math. SOC. Japan 36, 589-609, MR 85 # 460683). Especially, one derives that for a C*-algebra A the A-valued inner product of every Hilbert A-module A can be continued to an A-valued inner product on it's A-dual Banach A-module A' turning A' to a self-dual Hilbert A-module if and only if A is monotone complete (or, equivalently, additively complete) generalizing a result of M. HAMANA [Internat. J. Math. 3 (1992), 185-2041. A classification of countably generated self-dual Hilbert A-modules over monotone complete C*-algebras A is established. The set of all bounded module operators End,(&) on self-dual Hilbert A-modules dl over monotone complete C*-algebras A is proved again to be a monotone complete C*-algebra. Applying these results a WEYL-BERG type theorem is proved.

Recently, some progress was made in the theory of operator valued weights and conditional expectations of finite index on W*-algebras [3], and in a generalized TOMITA-TAKESAKI theory for embeddable A W *-algebras [12] using self-dual Hilbert W*-modules and Kaplansky-Hilbert modules as the essential technical tool. However, as almost everywhere in the literature, the theory was developed only as far as necessary for the solution of the main problem of the authors. Since the tool "Hilbert C*-module" has been used by more and more mathematicians, previously in operator theory, it might be quite useful to sharpen the tool for getting deeper results in it's areas of application. All the more there are only a few such well-known basic publications on this subject like [29], [40, 411, [43], [31], [33], [22], [32], for example. The present paper is an attempt to make a next step.

The concrete cause for the investigations below are two problems formulated by W. L. PASCHKE (cf. [40, 41]), and the problem of finding a purely C*-algebraical approach to the diagonalization of operators up to a suitable small rest (cf. [53, 541).

W. L. PASCHKE raised the question for which C*-algebras A the A-valued inner product on an arbitrary Hilbert A-module A! can be lifted to an A-valued inner product on its A-dual Banach A-module A' turning A' into a self-dual Hilbert A-module. By his

62 Math. Nachr. 175 (1995)

construction one can find such a lifting in the case of A being a W*-algebra or commutative AW*-algebra. On the other hand he showed that A has to be at least an AW*-algebra, cf. [40, Th.3.2.1, [41, Prop. 11. Refering to M. HAMANA [22, Th. 2.21 and H. LIN (33, Lemma 3.71 we resolve this problem for monotone complete C*-algebras affirmatively giving a general construction based on order convergence. Moreover, glueing together results of M. HAMANA [22] and K. SAITG, J. D. M. WRIGHT [44] we can show that this is the general solution of W. L. PASCHKE’S problem. However, the central problem of the A W *-theory, whether all AW*-algebras are monotone complete or not, is still open despite recent encouraging results ([7, 441).

The second problem under consideration is to describe the inner structure of self-dual Hilbert A-modules 4 over C*-algebras A. W. L. PASCHKE has found very nice criteria on self-duality and C*-reflexivity in the case of A being a W*-algebra or a commutative AW*-algebra. These ideas were extended by other authors. (Cf. [40], [13], [3]). Moreover, the author was able to show that the self-duality of an arbitrary Hilbert A-module A over a C*-algebra A does not depend on the properties of the concrete given A-valued inner product on A’ realizing the self-duality, but only depends on the existence of such an A-valued inner product on the Banach A-module 4 inducing an equivalent norm to the given one, [ 131. Therefore one has the possibility to characterize self-duality of arbitrary Hilbert C*-modules by their inner structure. We give such a characterization in the case of A being a monotone complete C*-algebra. On the way the interrelation is shown between the theory of Kaplansky-Hilbert modules and the theory of self-dual Hilbert A-modules over commutative A W*-algebras, cf. [37, Th.S.41, [9], [29], [39], [48].

These results allow us to prove some generalized WEYL-BERG type theorems for bounded, A-linear, normal operators on self-dual Hilbert A-modules with countably generated A-pre-dual Hilbert A-modules, where A is assumed to be monotone complete and to have a special approximation property (*). An example shows that a WEYL type decmposition can fail if the C*-algebra A does not have property (*).

Let us remark that the theory of operator valued weights and conditional expectations of finite index on W*-algebras in the approach of M. BAILLET, Y. DENIZEAU and J.-F. HAVET [3] can be generalized straightforward to the case of monotone complete C*-algebras using the results of the present paper. But this appears elsewhere, cf. [15]. The main content of the present paper was previously circulated as a preprint, [14].

The present paper is organized as follows: After a section about basic definitions and facts the second one is concerned with two

types of convergence in Hilbert A-modules over monotone complete C*-algebras A and their properties. In Section Three a spectral decomposition theorem for normal elements of normal A W*-algebras and a polar decomposition theorem for arbitrary A W*-algebras are proved. Two criteria on self-duality and C*-reflexivity of Hilbert C*-modules are established in Sections Four and Five. They have several consequences of interest: For every Hilbert A-module {A, (., .)} over a monotone complete C*-algebra A the A-valued inner product (., .) can be continued to an A-valued inner product on its A-dual Banach A-module 4’ turning A’ into a self-dual Hilbert A-module, (Section Four). One obtains a classification of self-dual, countably generated Hilbert A-modules over monotone complete C*-algebras A, (Section Six). The C*-algebra End,(A) of all bounded, A-linear operators on self-dual Hilbert A-modules A over monotone complete C*-algebras A turns out to be monotone complete, again. Hence spectral and polar decomposition are possible inside End,(.,&), and one can consider some kind cf VON NEUMANN representations of monotone

Frank. Hilbert C*-modules 63

complete C*-algebras on such Hilbert C*-modules (Section Seven). This last result gives us the chance to prove a generalized WEYL-BERG type theorem for the pair (End,(&), K,(&)} with A being a monotone complete C*-algebra with the approximation property (*). (Here K A ( A ) denotes the set of all bounded, “compact” operators on A.) This is explained in Section Seven.

1. Preliminaries and basic facts

Before we start considerations let us recall some definitions and fix the notations. We consider A W*-algebras, i.e., C*-algebras for which the following two conditions are satisfied,

(a) In the partially ordered set of projections every set of pairwise orthogonal projections

(b) Every maximal commutative self-adjoint subalgebra is generated by its projections,

([28l):

has a least upper bound.

i.e., it is equal to the smallest closed *-subalgebra containing its projections.

A C*-algebra is said to be monotone complete if and only if every bounded increasingly directed net {a,: a E I } of self-adjoint elements of it has a least upper bound a = sup {a,: a E I> in it. Every monotone complete C*-algebra is an AW*-algebra. At present it is unknown whether every A W*-algebra is monotone complete or not, cf. [9]. But commutative AW*-algebras are monotone complete, ([2]). More examples of monotone complete AW*-algebras can be found at [29, Th.7, Th.81, [40, Th. 11, [9]. An AW*-algebra is called normal if every increasingly directed net of projections { p a : a E I} with supremum p inside the net of all projections possesses a supremum with respect to the net of all self-adjoint elements (being equal to p automatically in the case of it’s existence), cf. [52]. A C*-algebra is additively complete if every norm-bounded sum of positive elements has a least upper bound.

If A is a monotone complete C*-algebra, then one can define order convergence on A. For the commutative case this was done by H. WIDOM [48] (cf. [l]), and for the general case by R. V. KADISON and G. K. PEDERSEN [24] who defined the so-called Kadison-Pedersen arrow. M. HAMANA [20, p. 2601 modified the latter notion to get a general notion of order convergence. A net (a, : a E I } of elements of A converges to an element a E A in order if and only if there are bounded nets {ap) : M E I } and {ba): a E I } of self-adjoint elements of A and self-adjoint elements a(k) E A, k = 1,2, 3,4, such that:

(i) 0 I a:”) - a(“) I b:“), k = 1, 2, 3,4, a E I ,

(ii) {bp) : a E I } is decreasingly directed and has greatest lower bound zero,

(iii) 4 4

(i)’ a:‘) = a, for every a E I , (i)k

We denote this type of convergence by LIM {u,: a E I } = a. By [19, p. 2601, the order limit of {aa: a E I > does not depend on the special choice of the nets {up ) : M E I } , {bkk): a E I } and of the elements a(k1, k = 1,2, 3,4. If A is a commutative AW*-algebra, then the order convergence defined above is equivalent to the classical order convergence in A as it was defined by H. WIDOM [48] earlier. Order convergence has the following properties, cf. [20, Lemma 1.21:

= a, (where i = m). k = 1 k = l

64 Math. Nachr. 175 (1995)

If LIM {a,: a E I} = a, LIM { b p : p E J } = b, then

(i) LIM {a, + b , , : a E I , ~ E J } = a + b,

(ii) LIM { xa,y : CI E I} = xay for every x, y E A,

(iii) LIM (a,bp: a E I , j3 E J ) = ab,

(iv) a, I b, for every CI E I implies a I b,

(v) llalla 5 lim SUP {IIa,IIA :ff E I}.

Throughout this paper we denote the C*-norm of A by I ) . I I A . The self-adjoint part of A is denoted by A, and the positive cone of A by A:. The centre of A has the denotation Z(A).

Now some facts about Hilbert C*-modules. We make the convention that all C*-modules of the present paper are left modules by definition. A pre-Hilbert A-module over a certain C*-algebra A is an A-module A' equipped with an A-valued, non-degenerate mapping (. , .) : A x A -+ A being A-linear in the first argument and conjugate-A-linear in the second, and satisfying (x, x) E A: for every x E A. The map (., .) is called the A-valued inner product on 22'. A pre-Hilbert A-module {.A, (., .)} is Hilbert if and only if it is complete with respect to the norm 1 1 . 1 1 = I/(., We always suppose that the linear structures of A and A are compatible. Denote by (A', A) the norm closed linear hull of the range of the map (., .) in A. A Hilbert A-module {A, (., .)} over a C*-algebra A is said to be self-dual if and only if every bounded module map r : 4 -+ A is of the form (., a,) for a certain element a, E A!'. The set of all bounded module maps r : 4 -+ A is denoted by A". It is a Banach A-module. A Hilbert A-module is said to be C*-reflexive (or A-reflexive) if and only if the map 52 being defined by the formula

Q(x) [r] = r ( x ) for each x E A' , every r E =A' ,

is a surjective module mapping of A onto the Banach A-module A'", where .A? consists of all bounded module maps from A' into A. A Hilbert C*-module A' is countably generated if there exists a countable set of elements of A the set of finite C*-linear combinations of which being norm-dense in A. For more basic facts about Hilbert C*-modules we refer to [40, 41, 321.

2. Two types of convergence

Definition 2.1. (Cf. [51, 9 21, [48, $ 1.11, [13, Def. 3.11, [l]). Let A be a monotone complete C*-algebra, {A', (., .)) be a pre-Hilbert A-module and I be a net. A norm-bounded set {x,:a E I } of elements of A' is fundamental in the sense of zy-convergence (in short: ry-jundamental) if and only if the limits

LIM {(x, - xp, X, - ~ p ) : E I}

exist for every fl E I, and the limit

LIM(LIM{(x, - x ~ , x , - x B ) : c I E I ) : / ~ E I }

Frank, Hilbert C*-modules 65

exists, too, and equals zero. A norm-bounded set { x , : a ~ l } of elements of A has the 1:-limit x in A if and only if the limit

LIM ((x, - X , X , - X ) M E I}

exists and equals zero. In this case one writes

7: - Iim { x , : a ~ I } = x ,

and one says that {x,: M E I } 7:-converges to x inside A.

Lemma 2.2. Let A be a monotone complete C*-algebra and {A, (., .)} a Hilbert A-module. I j{ x, : CI E I > is a norm-bounded set of elements of A indexed by a net I andpossessing a 7:-limit x in A, then the,following statements are true:

(i) The t:-limit x E Ji? is unique. (ii) The set ( x a : a E I } is z:-jundamental. (iii) If ~ ~ x , ~ ~ 5 N for every a E I and a real number N , then ~~x~~ 5 N . (iv) For every a E A the equality z: - lim {ax, a E I } = ax is satisfied. (v) For every norm-bounded set { y p : p E J } (not necessarily distinct f rom (x, : a E I } ) of

elements of dl indexed by N net J and possessing a 7:-limit y in d the equality 7: - lim {x, + yo : a E I, p E J } = x + y holds.

Proof. To prove the first fact suppose the existence of two 77-limits x l , x2 E A of the norm-bounded net {x, : M E I } . The inequality

0 I ( X I - X 2 , X I - x2) I 2((x, - x1 ,x , - X I ) + ( x , - x2 ,x , - xz))

holds for every a E I . If one calculates the order limit of the right side, one obtains the equality x1 = x2 . Furthermore, the fact (ii) can be derived from the inequality

0 I ( x , - x,, x , - xp) I 2((xa - x , x, - x ) + ( x s - x, x , - x ) )

being satisfied for every a, /3 E I, and from the 7:-convergence of the set { x a : M E I > to x E A. Similarly, (v) follows from the inequality

0 I (x + y - x , - Y , J , X + y - xa - YO)

I 2 ( ( x - x,, x - x,) + ( Y - Y,, Y - Y p ) )

being satisfied for every a E I , p E J , and from the 7:-convergence of the sets (x, : M E I } , { y , : p E J } to x, y E A, respectively. To show (iii) notice that by (v) there exists the order limit of the net {(x,, .x,) : a E I } in A being equal to ( x , x ) . Therefore

llxll = l l ( x , ~ ) I l ~ ’ ~ s l i m s u p { ~ l ( ~ , , x ~ ) I l ~ ’ ~ : ~ ~ I } I N .

Statement (iv) is derived from the fact that the inequality 0 5 ( x , x ) 5 ( y , y ) implies the inequality 0 I a(x, x ) a* I a(y , y ) a* for everyx, y E A, every a E A, ( [5 , Prop. 2.2.131). 0

Definition 2.3. (Cf. [48, 5 1.11, [13, Def. 3.11). Let A be a monotone complete C*-algebra and let {A, (., .)) be a pre-Hilbert A-module and I be a net. A norm-bounded set (x, : M E I> of elements of dL is fundamental in the sense of z$convergence (in short: z;TfundamentaI) if and only if the limits

LIM { ( y , x, - x a ) : a E I}

5 Math. Nachr.. Bd. I75

66 Math. Nachr. 175 (1995)

exist for every y E A, for every p E I , and the limit

LIM{LIM{(y,x, - x p ) : a ~ I } : f i ~ I }

exists, too, and equals zero. A norm-bounded set {x, : M E Z} of elements of A! has the @imit x in A if and only if for every y E .1 the limit

LIM ( ( y , x, - x): a E I )

exists and equals zero. In this case one writes

z2 0 - l i m { x , : a ~ I } = x ,

and one says that {xu: M E I ) . z:-converges to x inside -1.

Lemma 2.4. Let A he a monotone complete C*-algebra and {A, (., .)} be a Hilbert A-module. If {xu : M E I } is a norm-bounded set of elements of A indexed by a net I and possessing a z!-limit x in A, then the,following statements are true:

(i) The z;-limit x E A is unique. (ii) The set {x,: LX E Z} is z$jundamental.

(iii) I f I I . Y , ~ ~ I N for every LX E I and a real number N , then llxll 5 N . (iv) For every a E A the equality 7: - lim {ax, : a E I } = ax is satisfied. (v) For every norm-bounded set { y p p E J } (not necessarily distinct f rom {xu: M E I } ) of

elements of A indexed by a net J and possessing a z:-limit y in A, the. equality 2; - lim ( x u + yp : LX E I , f i E J } = x + y holds.

Proof . The facts (i), (ii) and (v) are obvious. To show (iv) one has only to consider the equality.

LIM { (y, ax,) : LX E Z} = LIM ( ( y , x,)a* : a E I }

= (LIM {(y, x.) : M E I}) a*

= (y,.x)a*

= (y,ax)

being valid for every a E A and every y E A. For the proof of (iii) consider the inequality

0 5 Il(X,,Y) + (Y,x,)ll 2 llYll llxll 5 2 llYll N

being valid for every y E A, every tl E i and some real positive number N by assumption. The element ((xu, y) + (y, xu)) is self-adjoint for every y E A, every M E I . Therefore

- 2 N IlYlI 1, 5 (Xm Y > + (Y, xu) 5 2N IlYll 1,

for every M E I and every y E 4. Replacing y by x and taking the order limit of the central expression statement (iii) turns out. 0

Lemma 2.5. Let A he a monotone complete C*-algebra, {A, (., .)} he a Hilbert A-module and Z be a net. IJ'a norm-bounded set { x u : M E I } of elements of A possesses a 7:-limit x in A, then that element x is the z:-limit of the net {xu: a E I ) in A, too.

Frank, Hilbert C*-modules 67

Proof . The statement above follows from the following equation based on the polariza- tion formula:

3

LIM {(x - x,, y) : M E I } = - LIM ik(x - x, + iky, x - x, + iky) 4 { k = O

1 3 = - C ik LIM {(x - X, + iky, x - X, + iky>}

4 k = O

= o being valid for i = p, every y E A and every a E I . Therefore x = 7; - lim {xu : c1 E I}. 0

For example, if one defines an A-valued inner product (., . ) A by the formula ( a , b ) := ab*, (a, b E A), on a monotone complete C*-algebra A, then the z;-convergence induces the order convergence on A. On the other hand, 7:-convergence induces the order convergence on A if and only if A is commutative. As another example one can consider an arbitrary Hilbert space X , (where A is the set of complex numbers). Then 7:-convergence is induced by the norm topology on 2, whereas the 7:-convergence is induced by the weak topology on 2, i.e., the two types of tO-convergence do not coincide, in general. Moreover, on Hilbert A-modules over W*-algebras A 7:-convergence is induced by a topology which is generated by the semi-norms {f((., .))''' :f E A*}, and 7;-convergence is induced by a topology which is generated by the linear functionals {f((., x)): f E A,, x E A}, cf. [13].

Remark. Does there exist a topology on monotone complete C*-algebras inducing order convergence on them? The answer is negative even in the commutative case. A simple example has been constructed by E. E. FLOYD [l 11 in 1955.

Proposition 2.6. Let A be a monotone complete C*-algebra and {A, (., .)} a Hilbert A-module. Consider the standard embedding of A into its A-dual Banach A-module A'. The linear hull of the 7:-completed unit ball of A 4 A' can be identified with a Hilbert A-submodule {A*, (., .),} of the Banach A-module A'. The embedding A 4 A' is realized via the mapping x E A + <., x), E A* since (x, y), = (x, y) for every x, y E &,

Proof. By Lemma 2.5 every 7:-fundamental set {xu: M E Z} of dl is z;-fundamental too. Embedding A into &' in the canonical way and defining

r (y ) = LIM { (y, x,) : c1 E Z}

for arbitrary elements y E A, one finds r = 7; - lim {x, : M E Z}. Let us denote the union of A c, A' with the set of all limits of it's ?:-fundamental subsets by A*.

The next step is to define an A-valued inner product on A* 5 A?". Set:

. (x,y)D = ( x , y ) , 4 A*, (x, r ) , = r ( x ) , x E A 4 A'*, r E A* / A.

This setting is correct because of the definition of r E A* / A' above. Moreover, since the inequality

0 5 r (y , - Y,)* r(y, - Ya) 5 llrllZ (Y, - Y6,Yy - Ya)

68 Math. Nachr. 175 (1995)

is valid for every fundamental set { y y : y E J } one defines

( ~ , r ) ~ = L I M { r ( y , ) : y € J } , r , s ~ . X * \ . . ~ , s = L l M { y , : y ~ J i .

Note that the sequence of the two order limits is irrelevant to the result. With such a setting define arbitrary values of the A-valued inner product with respect to the axioms of this structure. By Lemma 2.2,2.4 and 2.5 the A-valued inner product (. , . ) D is well-defined.

Now we have to show that the t$limit r of the ty-fundamental set {x,:c( E I] of .1 is its ~Y-limit, too. Obviously,

LIM {((x, - r , x , - r ) ) : c t ~ I ) = LIM { ( ( x , , x n - .xp)

- r (x , - x,)*): a, E I) = LIM { ( ( xa , xN - ~ p )

- ( x p , x, - x,)): a, B E 1) = 0 .

Hence every 7:-fundamental set of 4 has a 7:-limit inside A*. The set A?* is obviously complete with respect to the norm ll(., .)Dlli'z by its definition. 0

3. Spectral and polar decomposition inside A W*-algebras

We want to show a spectral theorem for normal elements of normal AW*-algebras. I t gives important informations about normal and, especially, self-adjoint elements being necessary for further considerations. To formulate the theorem the following definition is useful:

Definition 3.1. (Cf. [51, p. 2641). A measure m on a compact Hausdorff space X with values in the self-adjoint part of a monotone complete C*-algebra is called quasi-regular if and only if

m ( K ) = inf { m ( U ) : U-open sets in X , K c V )

for every closed set K c X . We remark that the condition

m(U) = sup { m ( K ) : K-closed sets, K c U ) for every open set U c X

is equivalent to quasi-regularity. Further, the measure m is called regular if and only if

m ( E ) = inf (m(U) : U-open sets in X , E 5 U )

for every Bore1 set E E X .

Theorem3.2. (Cf. [51, Th. 3.1, Th. 3.21. Let A he a normul AW*-ulgrbru und let U E A be u normul element. Let B E A be the commutative C*-suhulgrbru in A generated by the &ment.s ( I A , a, a*; , and denote by B the smallest cornmucutive AW*-algebra inside A conluining B and being monotone complete inside every maximal commutative C'*-subalgebra D of A with the property B c D. Then there exists unique quusi-regulur fi-oalued measure m

Frank. Hilbert C*-modules 69

on the spectrum o(a) c C of a E A, the values of which are projections in B and for which the integral

Adma = a a@)

exists in the sense of order convergence in B G A.

Proof. By GELFAND-NAIMARK the commutative C*-subalgebra B E A being generated by the elements {lA, a, a*} is *-isomorphic to the commutative C*-algebra C(o(a)) of all complex valued continuous functions on the spectrum a(a) c C of U E A . Denote this *-isomorphism by q, cp : C(o(a)) + B. The isomorphism cp is isometric and preserves order relations between self-adjoint elements and, hence, positivity of self-adjoint elements. Therefore q is a positive mapping.

Choosing an arbitrary maximal abelian C*-subalgebra D of A containing B one can complete B to B(D) with respect to order convergence inside D. But B(D) c A does not depend on the choice of D. This can be easily seen if one extends the map cp to an order preserving, isometric mapping @ from the set of all bounded complex-valued functions on o(a) into B(D) like in [50], [51]. The characteristic functions of the Borel sets of .(a) generate all projections of B(D) via 4. Moreover, every projection p of B(D) is the supremum of the net P = {a E B,: a I p } , (where the supremum is taken inside B(D)), [21, Lemma 1.71. It does not depend on the choice of D since P = P z and, hence, the supremum of P inside every other maximal commutative C*-algebra D' of A containing B is also a projection p'. The projections (1, - p ) and (1, - p') both annihilate P inside A. The latter implies p = p' because of the maximality of D and D' and of the normality of A.

Now, by [50, Th. 4.11, there exists a unique positive quasi-regular B-valued measure m with the property that

s f(4 dma = q(.f) a(a)

for every f E C(a(a)). Since q - ' ( a ) (A) = I for every I E o(a) c C, by the definition of q one gets

s I d m , = a . d U )

Moreover, since @ ( x ~ ) ~ = @(xi) = q?(xE), for the characteristic function of every Borel set E E .(a) the measure m is projection valued.

The following corollary is the key point for the subsequent considerations:

Corrollary 3.3. Let A be an AW*-algebra and {A,(., .)} be a pre-Hilbert A-module. I f x E A is diJjrerent f rom zero, then there exists a projection p E A:, p + 0, and an element a E A: such that a, p and (x, x>'I2 commute pairwise, and such that

a(x, X) l jZ = (ax, ax)'i2 = p .

Proof. Consider the commutative C*-subalgebra B of A generated by the elements { l,, (x, x)}. By Theorem 3.2 there exists a unique positive quasi-regular measure m on the Borel sets of a((x, x>'/') c R + being projection-valued in the monotone closure B(D) of B

70 Math. Nachr. 175 (1995)

with respect to an arbitrarily fixed, maximal commutative C*-subalgebra D of A which contains B, and satisfying the equality

f I dmn = ( x , u(<x, x) 1/21

in the sense of order convergence in B(D) c A. Now if ( x , x)’/’ is a projection set, a = l,, p = ( x , x ) . If ( x , x ) ’ / ~ is invertible in A set p = l,, a = ( x , x)’”. Otherwise consider a number ~ L E u ( ( x , x ) ” ~ ) , 0 < p < IIxII, and set K = [O,p] n CT((X ,X) ’ / ’ ) . The value m ( K ) E B(D) is a projection different from zero. It commutes with every spectral projection of ( x , x)lI2 and with ( x , x ) ’ / ~ itself. Since m is quasi-regular, one has

1 dm,(l, - m ( K ) ) = (1, - m ( K ) ) ( x , x)”’

Therefore one finds p = ( lA - m ( K ) ) and a = ((1, - m ( K ) ) ( x , x)-l /’ , where the inverse is taken in the C*-subalgebra (la - m(K)) B(D) c A. Since p < IIxII, the projection p is different from zero. The existence of a E A; is guaranted by 0 < p.

U ( ( X , X ) ~ / ~ ) \ X

0

For the completeness of the current section we show that a polar decomposition is possible inside every AW*-algebra. This generalizes assertions of S. K. BERBERIAN [4, 0 21, Prop. 1, Prop. 2, Exerc. 1,2], I. KAPLANSKY [30, Th. 651 and R. V. KADISON, G. K. PEDERSEN [24, Prop. 2.31.

Proposition 3.4 Let A be an A W *-algebra. For every x E A there exists u unique partial isometry u E Asuch that x = (xx*)li’ u andsuch that uu* is the rangrprojection of ( x ~ * ) ~ / ’ .

This follows from Corollary 3.3. and from the above cited results of S. K. BERBERIAN, I. KAPLANSKY and R. V. KADISON, G. K. PEDERSEN.

4. A criterion on self-duality and C*-reflexivity

Theorem 4.1. (Cf. [13, Th. 3.21). Let A be a monotone complete C*-algebra and {A, (., .)I he a Hilbert A-module. The following conditions are equivalent:

(i) A‘ is self-dual. (ii) J# is C*-reflexive. (iii) The unit ball of A is complete with respect to zy-convergence. (iv) The unit ball of .I is complete with respect to r$convergence.

If A is commutative there is a further equivalent condition, (cf. 137, Th. 5.41): (v) A? is a Kaplansky-Hilbert module over A.

Recall the definition of Kaplansky-Hilbert modules over commutative A W*-algebras:

Definition 4.2. [(29, p. 842, Def.]). Let A be a commutative AW*-algebra. A Hilbert A-module {A, (., .)> is Kaplansky-Hilbert if and only if it has the following two properties:

(i) Let { p a : CI E I } be a set of pairwise orthogonal projections of A with least upper bound p E A. Let x E J# be an element for which pax = 0 for every CI E I. Then px = 0.

(ii) Let { p a : CI E if be a net of pairwise orthogonal projections of A and let ( x u : CI E I> be

Frank, Hilbert C*-modules 71

any bounded set in A. Then there exists an element x E A such that pc,x = pax, for every c1 E I.

Before we start proving the theorem we make a simple observation:

Lemma 4.3. Let A be a monotone complete C*-algebra {A, (., .)} be a Hilbert A-module possessing a z$complete unit ball. r f f A --f A is an A-linear bounded mapping for which the set Ker (f)' = {x E A : (x, y) = 0 for every y E Ker ( f ) } consists only of the zero element, thenf = 0 on A.

Proof. Suppose that there exists an element y E A \ Ker cf) with the property f(y) + 0. One has to show that there exist elements p , a E A: for y E A with the properties described in Corollary 3.4 (i.e., p = (ay, ay), in particular), and with f(ay) += 0. Indeed, if f(ay) = 0 for every possible choice of p and a, then by the equality f(ay) = (pap) ( p f ( y ) ) = 0 and by the invertibility of a inside pAp one has p f ( y ) = 0. Consequently, f(y)* pf(y) = 0 for every possible chosen p E A:. But the supremum of all such projections p is the support of y E A (cf. Cor. 3.4), and by [20, Lemma 1.91 f(y) = 0 follows in contraction to our choice at the beginning. Now denote by L,(ay) the A-submodule of A generated by the element ay E A. Since L,(ay) is generated by a single element and norm-closed it is self-dual as a Hilbert A-submodule of A, ([35, Cor.]). Hence there exists a non-zero element z E L,(ay) such that

flLa(oy)(X) = (TZ)

for every x E L,(ay). But z E Ker (f)' in contradiction to the assumption.

Proof of the Theorem. First we prove the implication (iii) -+ (i). If the Hilbert A-module .D has a z;-complete unit ball, then we can suppose (A, A) = A without loss of generality. Indeed, (A, A') is a two-sided, monotone and norm-closed C*-ideal of A. Thus by [29, Cor. 2.3.11 a central projection p E A, p += 0, exists for which (A, A) = Ap. Therefore the Hilbert A-module A could be considered as a Hilbert Ap-module replacing A by Ap.

Consider a arbitrary A-linear, bounded mapping f : A + A. By Lemma 4.3 one can suppose Ker (f)' =k {O). The pair (Ker cf)', (., .>> defines a Hilbert A-submodule of A possessing a t;-complete unit ball. The map j is faithful on Ker (f)'. The image f(Ker (f) ' ) is a norm-closed left ideal I of A, and there exists a projection PEA: such that I = Ap. Indeed, since {Ker (f)', (., .)} and { I , (., .),} are isomorphic as Hilbert A-modules via f by assumption, one has

0

(f(y), f ( ~ ) ) ~ < I I f I I z (Y, y ) , (f-'(a),f-'(a)> I IIf-'I12 (a, a>,

for every a E I, y E Ker (f)', (because of the A-linearity off). But Ker (f)' has a t;-complete unit ball. Therefore the inequalities above yield I = Ap for a certain projection p E A: as desired since I \ f - ' l ] is bounded.

The Hilbert A-module {Ap, (., .),} is self-dual, and so is {Ker cf)', (., .)}. Conse- quently, there exists an element x E Ker cf)' such that f(.) = (., x) on Ker (f)'. Since (Ker ( f ) has a z;-complete unit ball, (i.e, Ker ( f ) = (Ker (f)')'), the Hilbert A-submodule .N = Ker ( f ) + Ker (f)' of A possesses a 7;-complete unit ball. The A-linear, bounded map g(.) = f(.) - <., x) is obviously equal to zero on N and N' = ( 0 ) 2 Ker (8)'. Therefore by Lemma 4.3 g = 0 on A, which proves the implication.

Now we show the implications (i) + (ii) + (iii). The first implications is trivial by definition so we concentrate our attention to the second one. Suppose that a C*-reflexive Hilbert

72 Math. Nachr. 175 (1995)

A-module {A, (., .)> does not have a 77-complete unit ball. Then consider the Hilbert A-module {.&*, (., .),) being derived from .A in the way described at Proposition 2.6. Since .L* is self-dual by the assertions above, since A 4 A* and since cdf’* c, A’, the Banach A-module A = A” has to be identical with A’* in contradiction to our assumption.

The conditions (i) and (iv) are obviously equivalent. Indeed, if a non-self-dual Hilbert A-module would possess a ti-complete unit ball, then its unit ball would have to be non-ry-complete by the previous observations. But this contradicts its .c’$completeness by Lemma 2.5. In the same manner one shows the inverse implication.

In the case of A being commutative the equivalence of (v) with some of conditions was shown by M. OZAWA [37]. For completeness, we give another proof using methods of order convergence. Let (p, : c( E I ) be a set of pairwise orthogonal projection of A possessing a least upper bond p E A. Let x E A such that p,x = 0 for every CI E I . Because of the inequality

0 = P , ( X , x> P, 5 P ( X 3 x> P

being valid for every CI E I , and because of the equality 0 = sup {p,(.u, x) p a ) = p(x , x) p being valid by (iii) and by the commutativity of A one has px = 0. So one has item (i) of Definition 4.2. Furthermore, let p = 1, and let {x, : CI E I } be a bounded set in A’ indexed by I . If 9 is the net of all finite subsets of I partially ordered by inclusion, then

and x has the properties to satisfy Definition 4.2, (ii) because of (iii). Therefore (iii) implies (v). That (v) yields (i) was proved by I. KAPLANSKY [29, Th. 51. So we are done. 0

Examples 4.4. (a) Let X be a stonean space and let C ( X ) = A be the A W*-algebra of all continuous, complex valued functions on X . Suppose that X consists of infinitely many points and, therefore, contains at least one accumulation point x E X . Let C,(X) be the set of all functions of C(X) vanishing at x. Obviously, C , ( X ) is an ideal in C ( X ) . Setting .L = C,(X) and A” = C ( X ) with the A-valued inner product (., .)a, one has .N’ = I hf” = JV’ = C ( X ) . That is, A is not self-dual.

(b) Let A be a monotone complete C*-algebra and consider the standard countably generated Hilbert A-module

1 Cn

I,(A) = a = (a, : i E N) :ai E A, a,$ is A-norm-convergent , I = 1

m

i (a ,b) = C aibT .

i = 1

It is self-dual if and only if A is finite dimensional as a linear space, ([13, Th. 4.31). The Banach A-module

turns into a self-dual Hilbert A-module if one defines the A-valued inner product by

Frank, Hilbert C*-modules 73

Corollary 4.5. Let A be a monotone complete C*-algebra and {A”, (., .)} be a self-dual Hilbert A-module such that a = 0 E A is the only element of A for which a Jke = { O ) . Then there exists an element z E A” with the property ( z , z ) = 1, and (A, A) = A.

Proof. One can choose a maximal set {x,:~ E I} of element of with respect to the conditions:

(i) (.xu, xu> = pU = P,Z =I 0 , (ii) paps = 0 for every CI =I p since the unit ball of A is 7:-complete, (cf. Corollary 3.3.

and ZORN’S lemma). Since the equality sup { p a : CI E I ) = 1, is valid, one can define

z = T: - Iim { C xi : Y E F} ie.Y

(where 9 is the net of all finite subsets of I), and one finds the desired element z E A”, and

{ ( a z , b z ) : a , b ~ A } = A G (A‘,JZ). 0

The following corollary generalizes a proposition of W. L. PASCHKE [40, Prop. 3.111, Corollary 3.3. and Proposition 3.4. It shows that one has something like polar decomposition inside self-dual Hilbert A-modules over monotone complete C*-algebras A.

Corollary 4.6. Let 4 be a self-dual Hilbert A-module over a monotone complete C*-algebra A. Every x E A” can be decomposed x = (x, x ) “ ~ u , where u E A! is such thul ( u , u ) is the range projection of (x, x)l/’. This decomposition is unique in the sense that if the equality x = bv is valid for b E Ah+, v E Jke such that (v , v ) is the range projection of b, then v = u and b = (x, x)’I2.

Proof. (Cf. [40, proof of Prop. 3.111). For a fixed x E A” one sets

x, = h, ‘x with h, = ((x, x) + l / n . 1,)’/’, ( n E N) . From the equality

(x,, x,) = ((x, x> + 1 / n . 1.4-l (x, x> one infers that [lx,Il I 1 for every n E N . Let Y E A be a r$accumulation point of the sequence { x, : n E N}. Since

x = h,x, , ( n E N) , and IIh, - (x, x)~/’(( -+ 0 for n + co

the equality (x, x ) l i2y = x follows. Denote by p the range projection of (x, x)l/’ in A. One has

(x, x)l /z ( P - P(Y3 y> P ) (x, x ) l l 2 = 0 .

(x, X ) l j 2 ( P - P ( Y , Y > dl” = 0

P ( P - P(Y> Y > P ) l i Z = 0 .

Since llyll I 1, the element ( p - p ( y , y ) p ) is positive. One obtains the equality

i.e.,

Hence p = p(y , y ) p and one sets u = p y . Then (x, x ) l iZ u = x and (u , u ) = pas desired.

74 Math. Nachr. 175 (1995)

To show the uniqueness of the decomposition suppose x = bv with b E A:, v E A such that ( v , v ) is the range projection of b. Then ( x , x ) = b2, and b = ( x , x ) ~ / ~ . Also (v, v ) = p. The equality ( v - pv , v - pv) = 0 forces v = pv. Also, (x, u ) = ( x , x ) l l 2 at one side and (x, u ) = ( x , x)’/’ (v , t i ) at the other side. Thus (x, x ) l i Z ( p - (v, u ) ) = 0, and

0 = P ( P - ( v , u ) ) = p - ( P V , U ) = p - ( u , u ) .

Hence ( u - v, u - v ) = 0 and u = v. 0 Now we get the following general solution of W. L. PASCHKE’S problem claimed in the

introduction:

Theorem 4.1. Let A he a C*-algebra. For every Hilvert A-module {A?, (., .)) the A-valued inner product (., .) on .I can be continued to a A-valued inner product (., .)n on the A-dual Banach A-module J.’ turning (A’, (., .),) into a seljldual Hilbert A-module !farid only if A is monotone complete (ilf A is additively complete). Moreover, the equalities

( x , Y > D = ( x , y > 9 ( x ? r > D = r ( x )

are satisfied for every x, y E A 4 A?, every r E A‘.

Proof. One direction follows immediately either from M. HAMANA [22, Th. 2.21 or from H. LIN [33, Lemma 3.71 or from Proposition 2.6 and Theorem 4.1 above. The converse can be seen combining the result of M. HAMANA [22] that the C*-algebra A has to be additively complete with the results of K. S A I T ~ and J. D. M. WRIGHT [44, (j 31 that additively complete C*-algebras are monotone complete, and vice versa. 0

Corollary4.8. Let A be a monotone complete C*-algebra and d be a self-dud Hilbert A-module. For every pre-Hilbert A-submodule N c .A? one can decompose -4 into the direct sum of .V* = N’ and (.N*)l = (N’)’.

This is a consequence of Proposition 2.6, Theorem 4.7, Theorem 4.1 and [13, Th. 2.81.

5. A structural criterion on self-duality and C*-reflexivity

In the present section we want to show a structural criterion. It was suggested by [40, Th. 3.111 and first proved by M. HAMANA f22, Th. 1.21, independently. We use our own methods to give another proof of it. To formulate the assertion we need the following definition:

Definition 5.1. Let I be an index set and let {(A,, (., .),} : a E I> be a set of pre-Hilbert A-modules over a fixed monotone complete C*-algebra A. Let 2F be the net of all finite subsets of I partially ordered by inclusion. Define the value ( x , y ) , ~ A for all I-tuples x = fxGl E .ka: M E I}, y = {y, E JH,: ct E I } and for every S E 9 by the formula

<x, Y>S = 1 (xi , Y i ) i . i S S

Frank, Hilbert C*-modules 75

Let A’ be the set of all I-tuples x = { x u E A,: a E I } for which the least upper bound sup {(x, x ) ~ : S E F} exists in A. Then define for x, y E A

(x, y) = LIM { ( x , y ) , : S E T} .

The linear space A is a (left) A-module with respect to the coordinatewise operations induced from the {A,: M E I } . Moreover, the mapping (., . ) :A x A --f A has all the properties of an A-valued inner product on A+%? by Lemma 2.2 and Lemma 2.5. Let us denote the pre-Hilbert A-module (A, (., .)> by T: - .F{A,: a E I > . Note that A is norm-complete (resp., self-dual) if and only if each A!,, a E I , is.

Theorem 5.2. Let A be a monotone complete C*-algebra and let {A, (., .)) be a Hilbert A-module. The .following two conditions are equivalent:

(i) A’ is self-dual. (ii) There exists a set { p a : c1 E I ) of not necessarily distinct projections of A such that A!’

and T: - C{Ap, : M E I) are isomorphic as Hilbert A-modules.

P r o of. The implication (ii) + (i) easily follows from Theorem 4.1 and from the self-duality

By Corollary 3.4. and ZORN’S lemma one finds a maximal set of elements of A+%?, { x u : M E I } , of the Hilbert A-modules {(Ap,, (., .),} : a E I } . There remains to show the converse.

with respect to the assumptions

(a) (x,,x,> = P, = pt t 0 ,

(b) (x, ,xp) = 0 if M += f l .

Let F be the net of all finite subsets of I being partially ordered by inclusion. The equality x, = p,x , is valid for every a E I , and thus one can define a mapping

T : A ! ’ + - r : - Z { A p U : a ~ I } , T ( x ) = { ( X , X ~ ) : E E ~ } .

This mapping T is obviously A-linear. To show the surjectivity of T consider

(a , pa : a E I} E T: - C{ Ap, : M E I } .

Define y, = (aixi : i E S} E A for every S E F and y = 7; - lim {ys: S E F} , (cf. Th. 4.1).

One has y E A’ and T(y) = (anpa: a E I ) . To prove that T is one-to-one suppose the existence of a non-zero element x E A for which (x, x u ) = 0 holds for every a E I . By Corollary 3.4 the equality (ax , a x ) = p = p 2 t 0 holds for some p E A,,, a E A;. Moreover, since {ax , x u ) = 0 for every M E I and since the set {xu : M E I > is chosen to be maximal with respect to (a), (b) one gets ax = 0 and ax = 0 in contradiction to the choise of p . Finally, one has to show that (T(x ) , T ( x ) ) = (x, x) holds for every x E A. Indeed, for every x E A and every S E .F,

<xs> xs> = C (x, xi> pi(xi, x) = (~(x,), ~ ( x s ) ) i e S

where xs = Z((x, xi) x i : i E S} by definition. The desired equality now obtains by taking the 7:-limit on both sides of this equality. 0

76 Math. Nachr. 175 (1995)

6. Applications

First we formulate a classification of self-dual (and hence C*-reflexive), countably generated Hilbert C*-modules over monotone complete C*-algebras.

Theorem 6.1. Let A be a monotone complete C*-algehra and let .& be a seifidual, countably generatrd Hilbert A-module. Then there exist only the,following two possibilitie.s,for the inner structure of A and of A:

(i) dt! is finitely generuted and A is arbitrary. (ii) At' is decomposable into the direct sum of a finitely generated Hilbert A-module and

a countably generated Hilbert B-module, where B is a finite-dimensionul, two-sided C*-ideal of A.

Proof . The first statement follows from [35, Cor.]. One has to show the second one and the completeness of the classification. Suppose that the Hilbert A-module M is self-dual and countably generated. By Theorem 5.2 one has

A = T: - C{Api : i E N}

for a certain countable set { p i : i E N) of projections of A. By an inductive process (dividing in direct summands and taking direct sums), one can reach a situation in which the pairwise product p a p j = r of every two projections pipj ( i < j ) of our choice is a projection if and only if r = pi + 0. Suppose that the situation is realized. Since A! is countably generated, the sequence

1 u p ; : N E N

has to converge with respect to the A-norm for every element a = {a , sApi , EN} of T: - C(Ap, : icN) . Therefore if there are more than a finite number of infinite dimensional, two-sided A W*-ideals (Ap,, Ap,) of our choice, then the Hilbert A-module 7; - CCAp,: i E N) cannot be countably generated, cf. [13, Th. 4.31. Moreover, if there does not exist a finite dimensional, two-sided C*-ideal B in A containing all the finite dimensional, two-sided C*-ideals (Ap,, Ap,) of our choice, then the Hilbert A-module T: - C(Ap,: i E N ) cannot be countably generated by [13, Th. 4.31, again. So the statements follow. 0

Secondly, extending [40, Th. 3.71 we show how A-linear, bounded operators on a Hilbert A-module J%' over a monotone complete C*-algebra A can be continued to A-linear, bounded operators on the A-dual Hilbert A-module .A' in unique way.

Proposition 6.2. Let A be u monotone complete C'-algebra and let {A, (., .)} be ( I Hilbert A-module. Then every A-linear bounded operator T : dt! -+ A can be continued to a unique A-linear bounded operator T' : A' + A? on the A-dual Banach A-module A' of',,& preserving the operator norm. Moreover, if the operator T has a adjoint operator T* ;.A + .A', then (T*)' = (T')*.

Proof . By [40, Th. 2.81 one obtains

(1) (W), T ( x ) ) 5 IITII (x, x>

Frank. Hilbert C*-modules 77

for every x E A?. Hence one can define the operator T‘ : 4‘ + &‘ by the formulae:

T’(x) = T ( x ) for every x E &,

T’(zY - lim {x, : tl E I } ) = 7: - lim { T(x,) : tl E I> ,

where {x, : tl E I} is an arbitrary chosen 7:-fundamental net of A. Obviously, the operator T’ is A-linear and bounded by IlTIl, cf. Lemma 2.2 (iv) and (1). It is unique by (l), and by Lemma 2.5 one has (T*)’ = (T‘)*. 0

The following corollary generalizes results of G. WITTSTOCK [49]:

Corollary 6.3. Let A be a monotone complete C*-algebra, let {A, (., .)} be a self-dual Hilbert A-module and JV be a Hilbert A-submodule of &. Then every A-linear bounded operalor T : .N -+ & can be continued to a unique A-linear bounded operator T‘ : A + & preserving the operator norm and the relation T’(M’) = (0). The self-duality of & is necessary, in general, ,for the result.

Proof. Consider the unique extension T’ :MI + &’ with jlT’ll = IITlj existing by the previous proposition. Since 4’ = N’ @ M’, one defines the final operator on A by T‘ on M‘ and by the zero operator on M’.

To show that the restriction on & to be self-dual can not be dropped, in general, one constructs a counterexample. Consider a norm-closed left ideal D of A, where D is order dense in A and unequal to A. (For example, take the set of all bounded linear operators on a separable Hilbert space as A and the compact one’s as D.) Set A = A @ D and M = D @ D with the usual inner products on them. Note that M is contained in A?‘ as a submodule. But the operator T : (dl, d,) -+ (0, d , ) can not be continued to an operator T’ in any way. 0.

Thirdly let & be a self-dual Hilbert A-module over a monotone complete C*-algebra A. Denote by End,(&) the set of all bounded A-linear operators on A. We consider C*-subalgebras M of End,(&) coinciding with their bicommutant M” inside End,(&). The following fact was obtained by M. HAMANA [22, Prop. 1.21 independently:

Theorem 6.4. Let A be u monotone complete C*-algebra and let A be a selfdual Hilbert A-module. Then every C*-subalgebra M of End,(&) which coincides with its bicommutant M” inside End,(A) is a monotone complete C*-algebra.

P r o of. Consider an arbitrary bounded, increasingly directed net { B, : M E I } of positive elements of M. By [34, Th. 31 an element C E End,(&) is positive if and only if (C(x), x) 2 0 for every x E A. Therefore by Theorem 4.1 there exist an element B E End,(&) being defined by the formula

B(x) = 7’: - lim {B,(x): tl E I } , x E ,

and B = sup [B,:a E I} in End,(A). But for every B,, CI E I, and every element C of the commutant M’ of M the equality CB, - B,C = 0 is valid. Since

BC - CB = LIM{(CB, - B , C ) : ~ E I } = 0

inside End,(&), one has B E M” = M. 0

78 Math. Nachr. 175 (1995)

This result seems to be of some importance. Immediatelly one realizes that polar decomposition and spectral decomposition work inside M” = M c End,(A) in every such case, (cf. Theorem 3.2 and 3.6). For investigations about similar monotone complete C*-algebras Theorem 6.4 allows, for example, to introduce a notion of “Morita equivalence in order” along the line of the ideas of M. A. RIEFFEL [43] for W*-algebras. Another area of application is the theory of operator valued weights and conditional expectations of finite index between monotone complete C*-algebras following M. BAILLET, Y. DENIZEAU and J.-F. HAVET [3]. But these investigations appear elsewhere, cf. [15]. What we will do is to prove a generalized WEYL-BERG theorem, which will finish up the present paper.

7. A Weyl-Berg type theorem

The problem of approximation of normal elements in C*-algebras A by diagonalizable elements of A up to a suitable small remainder is part of the sphere of interests of many mathematicians. For an overview on the recent results and open problems compare the papers of D. VOICULESCU [47], R. V. KADISON [25, 26, 271, K. GROVE and G. K. PEDERSEN [16], G. J. MURPHY [36], N. HIGSON and M. R0RDAM [23], L. G. BROWN and G. K. PEDERSEN (61 and S . ZHANG [53, 541. What we would like is to show a WEYL-BERG type theorem for monotone complete C*-algebras and some corollaries of it.

Definition 7.1. A monotone complete C*-algebra A has the approximation property (*) if there exists a chain of pairwise orthogonal projections { p , : c1 E I } of A, with least upper bound 1, such that for every CI E I the monotone complete C*-algebra p,Ap, possesses a faithful state f, with the property that the norm-completion of the pre-Hilbert space (p.Ap,, f,(L .)),I is separable.

Note that in the case of A being a W*-algebra the states f , can be chosen as normal states.

The class of monotone complete C*-algebras with the approximation property (*) is sufficiently large to contain most of the W*-algebras of physical interest. The DIXMIER algebra D([O, 11) has property (*), too. But there are remarkable examples of W*-algebras which do not have property (*), cf. [5, Remark after Def. 2.5.1.1. Other examples of commutative AW*-algebras without property (*) were constructed by M. OZAWA [38] which can be seen by comparing Proposition 7.6 and Theorem 7.3 below.

Let us denote the norm-closure of the linear hull of

{ O x , y E End,(A):OXJz) = (z, x) y for every z E A, each x, y E *A}

by K A ( A ) as usually. K A ( A ) is called the set of “compact” operators on A’.

Definition 7.2. Let A be a monotone complete C*-algebra, let A be a self-dual Hilbert A-module possessing a countably generated A-pre dual Hilbert A-module. Let 0 .t. q = q2 E A,, be fixed. An operator D E qEnd,(A’) is said to be diagonalizahle if there is

a sequence {xn: n E N) of pairwise orthogonal elements of q A such that D(xn) = a,x,

for some {a,) E qAq in the sense of Ty-convergence inside q A and the subsets {ax, : a E A}, ( n E N}, are norm-closed.

m

n = l

Frank, Hilbert C*-modules 79

Note that in the case of A being commutative, A can be identified with the centre of End,(&) and, hence, D is diagonalizable inside qEnd,(&). So we are in the classical situation.

Theorem 1.3. Let A be a monotone complete C*-algebra with property (*). Let A+’ be a self-dual Hilbert A-module possessing a countable generated A-pre-dual Hilbert A-module. Suppose that qEnd,(&) * &,(A) for every central projection q E A. Then for every E > 0 and every self-adjoint operator T E End,(A) there exist a diagonalizable, self-adjoint operator D E End,(=&’) and a “compact”, self-adjoint operator K E K A ( A ) such that T = D + K and

If qEnd,(=&‘) = qK,(A) for a central projection q E A and if A is a W*-algebra or a commutative AW*-algebra, then the Hilbert A-module A is finitely generated and every self-adjoint bounded module operator is diagonalizable.

IlKll < c.

Proof. The C*-algebra A has property (*), and [36, Th. 91 is valid. Consequently for every E > 0 and every c1 E I there exist a diagonalizable, self-adjoint operator D, E p,End,(A) with eigenvalues a,, E (11, : A E C} and a “compact”, self-adjoint operator K , E puKa(&) with IlrC,lI < E such that p,T = D, + K,.

Since these operators {D,}, {K,} are pairwise orthogonal, linear operators on A, one can sum them up in the sense of order convergence inside the monotone complete C*-algebra

( A + EndA(&)) c End,(&).

One gets

T = ~ p , T = ~ D , + ~ K , = D + K C l C I a e r

where D E End,(&) is diagonalizable and self-adjoint, and K E K A ( A ) is “compact” and self-adjoint by construction. Moreover, IlKll < E.

If qEndA(A) = qK,(&) for a central projection q E A , then the Hilbert A-module q - 4 is algebraicly finitely generated by [45, Remark 4.51, [lo, Prop. 3.21. Since every normal element of M,(A) is diagonalizable for every n E N and for A being a W*-algebra or a commutative AW*-algebra by [26, 71, and since q(&) = P(qA”) for a natural number n and an A-linear projection P on A”, the desired result yields. 0

To extend the statement of Theorem 7.3 to normal elements T E End,(&) note that there always exists a self-adjoint element S E End,(&) such that Tis contained in the C*-subalgebra being generated by S and the identity of End,(&). This follows from Theorem 3.1 by functional calculus, (cf. [17, 18,461). Now the techniques of G. J. MURPHY [36, p. 2831 allow to prove the following:

Theorem 1.4. Let A be a monotone complete C*-algebra with property (*). Let =&’ be a self-dual Hilbert A-module possessing a countably generated A-pre-dual Hilbert A-module. Suppose that qEnd,(A) =l qK,(&) for every centralprojection q E A. Then for every normal operator T E End,(&) there exist a diagonalizable operator D E End,(&) and a “compact” operator K E End,(A) such that T = D + K .

If qEnd,(&) = qK,(A) for a central projection q E A and i f A is a W*-algebra or a commutative A W *-algebra, then the Hilbert A-module q& is finitely generated and every normal bounded module operator is diagonalizable.

80 Math. Nachr. 175 (1995)

Remark 7.5. The property of AV to possess a countably generated Hilbcrt A-module as it’s A-pre-dual cannot be dropped, cf. [18]. But the infinite cardinality of minimal generator sets of certain A-pre-dual Hilbert A-modules of A?’ has to be unique in some sense as shown below in Proposition 7.6. Unfortunally, it is not quite clear at present if A has to possess property (*) to validate the statement of Theorem 7.4, or whether other situations are possible. Beside this it is strange that a property of A should determine the truth of the generalized WEYL-BERG theorem with respect to the pair (End,(.l), K,(.&)}, because the intersection of the three C*-algebras inside End,(A) is only Z(A), the centre of A. So the translation of (*) to a C*-condition on (End,(A), KA(&)) appears to be non-easy if one tries to formulate it without reference to A, (at least if A is non-commutative). However, there are some facts that shead light on the situation, cf. Proposition 7.8.

To prove the following fact one makes use of a discovery of M. OZAWA [38]. He showed that the following misbehaviour may happen: There are special commutative A W*-algebras A such that the self-dual Hilbert A-module E,(A)’ is isomorphic to the self-dual Hilbert A-module 7: - C(A,,, : CI E I > for a pre defined set I of uncountable cardinality card ( I ) , where card (I) depends on the inner structure of A.

Proposition 1.6. Lct A be a commutative AW*-ulgebru. V’Jbr a given real number c > 0 one can decompose every selfladjoint operator T E End,(l,(A)‘) into the sum T = D + K of’ u diugonufizable, self-adjoint opwator D E End,(t,(A‘)) and a “compact”, self-utijoint operator K E KA(/2(A)’) with IlrClI < I:, then the existence of’ an A-linear isomorphism of’ /,(A)’ to a .s~l fduul Hilhert A-module of type T: - C{A,,,: M E I ) ,for sets I qf uncountuhlc curdinality card ( I ) is impossible, i.e., the structure o j A cannot be arbitrary.

Proof. Suppose that /,(A)’ is isomorphic to T: - C{A,,: CI E I } for a set I of uncoun- table cardinality card ( I ) . Denote by 2 the non-separable Hilbert space 7: - C{C,,,:a E I ) . By [18] there exists a self-adjoint operator To E Endc(%‘) being not decomposable into the sum of a diagonalizable, self-adjoint operator and a compact, self-adjoint operator inside Endc(X). Since

E,(A) = 7’: - CiA,,): CI E I > = (A @ %‘)’

by assumption, one can define a self-adjoint, bounded, A-linear operator T : (A @ 2)’ + (A 0 X)’ as the unique extension of the mapping

T’:a @ h + a @ To@), ( u E A , h~ X’ ) ,

from A @ %‘ to (A 0 .#)‘, (cf. Prop. 6.2). Assume now that T would be decomposable into the sum of a diagonalizable, self-adjoint operators D E End,(I,(A)’) and a “compact”, self-adjoint operator K E KA(12(A)’). If one considers A as C ( X ) for a certain compact Hausdorff space X , one finds

7;x) = To = D(X) + K,X)

with D,x) E Endc(HX) - diagonalizable and self-adjoint, K(,) E Kc(HX) - compact and self-adjoint. But this contradicts the choice of To. 0

Corollary 1.7. There are commutative A W*-algebras A such that every normal element of‘ M,(A), (n E N - arbitrary), is diagonalizable, hut the generalized WEYL-BERG theorem is not valid for all A-linear, hounded, self-ug’joinl operators on I, (A)’.

For an example compare [38], [16, Added in proofj and Proposition 7.5.

Frank, Hilbert C*-modules 81

Proposition 7.8. Let A be a monotone complete C*-algebra possessing in countable chain

{q,: k E N} of pairwise orthogonal projections such that 1 q, = 1, in the sense of order

convergence, and q, - 1, for every k E N. Let & be a self-dual Hilbert A-module with a countably generated A-pre-dual Hilbert A-module. Then End,(&) = K A ( A ) .

m

k = 1

Proof . By Theorem 5.2 we have & = T: - C{Ap,:n E N ) for projections { p n ) E A. Denote by u, the partial isometries of A realizing the equivalences q, - l,, i.e., ukul = qk, uluk = 1,. Then

id,(.) = (., u ) u = e , , , ( . ) where u = { u k p , : k E N} E .1. That is, the identity operator on A is “compact”. Since KA(&) is a two-sided ideal in End,(&), the proof is complete. 0

Similarly, if A! is finitely generated then End,(&) = K , ( . k ) , too, for every unital C*-algebra A. Especially, if A is a W*-algebra or a commutative AW*-algebra, then every element of End,(A) is diagonalizable, cf. [25, 261 and [8, Cor. 3.31. Hence the following problem is unsolved:

Problem. Is every normal bounded module operator on finitely generated Hilbert A-modules over monotone complete C*-algebras A diagonalizable?

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Universitat Leipzig FB Mathematikllnformatik Mathematisches Institut 04109 Leipzig Augustusplatz 10 Federal Republic of Germany frank@mathematik.uni-leipzig.d4OO.de