Math. Log. Quart. 45 (1999) 2, 203 - 206

Mathematical Logic Quarterly

@ WILEY-VCH Verlag Berlin GmbH 1999

Weak-operator Continuity and the Existence of Adjoints

Douglas Bridges and Luminita Dediu

Department of Mathematics, The University of Waikato, Hamilton, New Zealand

Abstract. It is shown, within constructive mathematics, that the unit ball &(H) of the set of bounded operators on a Hilbert space H is weak-operator totally bounded. This result is then used to prove that the weak-operator continuity of the mapping T w AT on &(H) is equivalent to the existence of the adjoint of A.

Mathematics Subject Classification: 03F60, 25340, 46830, 47530.

Keywords: Constructive mathematics, Weak-operator continuity, Weak-operator totally bounded, Limited principle of omniscience, Existence of adjoints.

1 The topology T~ on B ( H )

Let H be a Hilbert space, B ( H ) the space of bounded linear operators on H , and & ( H ) the unit ball of B ( H ) . The weal-operator topology T~ on B ( H ) is the weakest topology with respect to which the mapping T - (T+,y) is continuous for all z, y E H . The following results can be proved using classical logic, and are well known (see [8, pp. 304-3061).

1. T~ is determined by the seminorms T - J(Tz,y)j, where 2, y run through a dense subset of B1 ( H ) .

2. & ( H ) is Tw-compact. 3. For each A E B ( H ) the mappings T - T A and T - AT of ( B ( H ) , T ~ ) into

( B ( H ) , T ~ ) are continuous.

We are interested in the constructive properties of rw: those that can be estab- Iished using intuitionistic logic, a logic that leads to a wider range of models for the resulting mathematics. Theorems proved with intuitionistic logic not only have the normal classical interpretation, but also can be interpreted within BROUWER’S intu- itionism [lo], recursive mathematics [9], and, we believe, all current frameworks for computable analysis (such as WEIHRAUCH’S theory of Type I1 Effectivity 1111). We assume that the reader has access to [l] or [2] for the basic constructive theory of metric, normed, and Hilbert spaces.

Of the three classical results mentioned above, the first certainly holds construc- tively. The second almost does: & ( H ) is .rw-totally bounded (see Theorem 1 below); but, as is shown in [3], the .r,-completeness of & ( l 2 ) implies the Limited Principle of

204 Douglas Bridges and Luminiia Dediu

Omniscience (LPO), If (a,) is Q binary sequence, then either a, = 0 for all n or else there ezists n such that Can = 1,

which cannot be derived in intuitionistic logic and which is false in the recursive model (see [4, Chs. 2 and 71).

The proof of the continuity of the mapping T - TA on B ( H ) is relatively trivial, both classically and constructively. Classically, it is an immediate consequence of the compactness of & ( H ) that T - TA is uniformly continuous as a mapping of (&(H), T,) into (L?(H), T,). Constructively, we must do a little more work (see [S]), since we cannot prove the uniform continuity theorem (see [4, Ch. S]), and even if we could, we only know that & ( H ) is totally bounded.

The classical proof of the continuity of left multiplication with respect to the weak-operator topology is a trivial consequence of the identity (ATz, y) = ( T z , A*y) (z, y E H ) and the uniform continuity of the mapping T - (Tz ,z ) on Bl(H) for all 2, z in H . Construct,ively, this proof is fine when A* exists, but will not work in general as the statement, “Every element of B ( H ) has an adjoint” implies LPO.

At this point one might ask, “What is the problem, constructively, with the clas- sical method of obtaining A*y for any A E B ( H ) and y E H : namely, apply the Riesz Representation Theorem to the bounded linear functional t - ( A t , y) ?”. In order to apply the Riesz Representation Theorem constructively to a linear functional f on H , we need to know that f is not just bounded but has a norm, in the sense that sup{[ f(z)I : t E H , (It(( < 1) exists (see [2, p. 419, (2.3)]); since the classical least upper bound principle does not hold constructively (it entails LPO), we may not be able to find the supremum in question. However, the classical proof of the existence of A*y will work for us if we know that sup{((Az,y)( : y E H , lly(( < 1) exists for each y E H .

We now show, by a Brouwerian example’) that we cannot hope to prove construc- tively even the sequential continuity of the mapping T - AT at 0 with respect to the weak-operator topology.

Let (en) be the usual orthonormal basis of unit vectors in the Hilbert space 12, and for each positive integer n define T, E & ( H ) such that Tnel = en and, for k 2 2, T,ek = 0. Then T, has an adjoint, and the sequence (T,) is weak-operator convergent to 0. Now let (a,) be a binary sequence with at most one term equal to 1, and define A E &(H) by Az = ( c,“=l a,(z, e,))el. Suppose that the mapping T c, AT of (&(H),T,) into ( B ( H ) , T , ) is sequentially continuous at 0 - that is, maps se- quences converging to 0 to sequences converging to 0. Then there exists N such that a, = I(ATnel, e l ) ( < 1, and thus a, = 0, for all n 2 N. By testing a l , . . . , U N - ~ , we can therefore prove that Vn(a, = 0) V %(a, = 1). Hence the proposition

“For each A E B ( H ) the mapping T - AT of (&(H), 7,) into ( B ( H ) , T ~ ) is sequentially continuous at 0 ”

implies LPO. It follows that if A* exists for each A E B ( H ) , then LPO holds. (This is shown directly in [5].)

])For more on Brouwerian examples, see [4, Ch. 11,

Weak-operator Continuity and the Existence of Adjoints 205

This example raises the (classically vacuous) question:

“If, f o r a given element A of B ( H ) , the mapping T - AT of ( B l ( H ) , T ~ )

into ( B ( H ) , T ~ ) is continuous - or, in this case equivalently, uniformly con- tinuous - does A’ exist ?”

We show in Theorem 2 that, even the preservation of total boundedness (a conse- quence of uniform continuity) of the mapping in question is equivalent to the existence of A*.

2 The main results

We first prove the r,-total boundedness of B1 ( H ) in the general case, without assum- ing the separability of H .

T h e o r e m 1. Let H be a Hilbert space. Then B l ( H ) is weak-operator totally bounded.

P r o o f . Let E > 0. By [2, Lemma 2.5 on p. 3081, there exists a finite dimensional subspace Ho of H such that - Pzi(l < ~ / 5 for 1 < i < m, where P is the projection of H onto Ho. Since Ho is finite dimensional, each element of B(H0) has a norm, and B(H0) is a finite-dimensional Banach space, and therefore has a compact unit ball &(Ho), with respect to the usual operator norm. Let {Tf , . . . ,T:} be an &-approximation to Bl (Ho)> and consider any T E B l ( H ) . The restriction (PT)o of PT to HO belongs to Bl(Ho) , and hence there exists m with 1 6 m < r such that

II(PT)o - TAlI = sup{llPTz - T:xll : z E Ho, 11211 < 1) < E/5

For any vectors 7 in the unit ball of H we now have

I((T-TPK177)K l ( ( ~ - ~ ~ ~ ) t 1 ~ 7 7 ) l + I ( ( ~ - ~ ~ ) € , ~ - Pdl = I V T - T P ) E I v)l + 211t1111v - P711 < l((PTP - T$P)<, 7)l + l(PT - T P ) ( t - w 1 v)l + 2&/5 < II(PT)o - T$ll+ 2llE - PEll + 2&/5 < (c/5) + ( 2 ~ / 5 ) + (2&/5) = E .

Hence the finite set {T fP , . , . T:P} is a weak-operator &-approximation to &(IT) . 0

T h e o r e m 2. Let H be a Hilbert space, let A E B ( H ) , and let fA be the linear mapping T - AT of ( B l ( H ) , rw) into ( B ( H ) , rw). Then the following conditions are equivalent:

(a) fA is continuous at 0.

(b) fA is Tw-uniformiy continuous on & ( H ) . (c) fA maps every totally bounded subset of ( B l ( H ) , rw) t o a totally bounded subset

(d) A has an adjoint. of ( B ( H ) , r w ) .

P r o o f . To show that (a) j (b) is routine. The implication (b) 3 (c) is a special case of the general result that uniform continuity preserves total boundedness. That (d) j (a) we have already noted. Therefore it suffices to prove that (c) 3 (d).

206 Douglas Bridges and Luminifa Dediu

Assuming (c), fix y and a unit vector e in H . For each x in the unit ball B of H define T, E & ( H ) such that T,e = x and T,z = 0 for all z I e . Since Te E B for each T E & ( H ) , we see that B = { Te : T E B1 ( H ) } and therefore

A AX,^) : x E B } = {(ATe,y) : T E B l ( H ) } . So, in order to apply the Riesz Representation Theorem to construct A*y, it suffices to show that the set C = {I(ATe, y)I : T E & ( H ) } has a supremum in R. As & ( H ) is weak-operator totally bounded, we see from (c) that {AT : T E & ( H ) } is also weak-operator totally bounded. The uniform continuity of the mapping S - (Se, y ) on norm-bounded subsets of B ( H ) now ensures that the set C is totally bounded, and

0 therefore has a supremum in R. For related constructive results on the existence of adjoints, see [5].

References

[I] BISHOP, E., Foundabions of Constructive Analysis. McGraw-Hill, New York 1967. [2] BISHOP, E., and D. BRIDGES, Constructive Analysis. Springer-Verlag, Berlin-Heidel-

berg-New York 1985. [3] BRIDGES, D. , On weak operator compactness of the unit ball of L ( H ) . Zeitschrift

Math. Logik Grundlagen Math. 24 (1978), 493 - 494. 141 BRIDGES, D., and F. RICHMAN, Varieties of Constructive Mathematics. London

Math. SOC. Lecture Notes 97 (1987). [S] BRIDGES, D. , F. RICHMAN, and P . SCHUSTER, Adjoints, absolute values, and polar

decompositions. Submitted. [6] DEDIU, L., Constructive theory of von Neumann algebras. University of Waikato,

D. Phil. thesis. In preparation. [7] ISHIHARA, H. , Continuity and nondiscontinuity in constructive mathematics. J. Sym-

bolic Logic 56 (1991), 1349 - 1354. [8] KADISON, R. V., and J. R. RINGROSE, Fundamentals of the Theory of Operator Alge-

bras, Vol. 1. Academic Press, New York 1983. [9] KUSHNER, B. A., Lectures on Constructive Mathematical Analysis. Amer. Math. SOC.,

Providence, RI, 1985. [lo] TROELSTRA, A. S., and D. VAN DALEN, Constructivity in Mathematics. 2 Vols. North-

Holland Publ. Comp., Amsterdam 1988. [I 11 WEIHRAUCH, K ., A foundation for computable analysis. In: Combinatorics, Complexity,

& Logic, Proceedings of Conference in Auckland, 9-13 December 1996 (D. S. BRIDGES, C. S. CALUDE, J. GIBBONS, S. REEVES, I. H. WITTEN, eds.), Springer-Verlag, Singa- pore 1996, pp. 67 - 89.

(Received: February 17, 1998)