A Reflexivity Theorem for Weakly Closed Subspaces of Operators

It was proved in [4] that the ultraweakly closed algebras generated by certain contractions on Hubert space have a remarkable property. This property, in conjunction with the fact that these algebras are isomorphic to Hx, was used in [3] to show that such ultraweakly closed algebras are reflexive. In the present paper we prove an analogous result that does not require isomorphism with Hx, and applies even to linear spaces of operators. Our result contains the reflexivity theorems of [3, 2 and 9] as particular cases. Let áC(Jf) denote the algebra of (linear, bounded) operators acting on the Hubert spaced, and let Jldenote a linear subspace of ££( J?). Then Jf is endowed with the weak and ultraweak topologies that it inherits from £^(3^) (cf. [6, §15]). For two arbitrary vectors x, y g jFwe can define the (ultra) weakly continuous functional [x ® y] on^by [x®y](A)=(Ax,y), A e Jf, where ( • , • ) stands for the scalar product in Jif. Definition 1. Let « be a natural number, n > 1. The subspace Jf has property (B„) [respectively (A„)] if for every positive number e there exists a positive number S = 8(e, n) such that for every system {^¡f. 1 < /, / < n) of weakly [respectively ultraweakly] continuous functionals on J(and every system {x„ y,: 1 < /', j < n) of vectors in Jf satisfying the inequalities ||,-[x, ® yj]\\ < S there exist vectors {x¡, yj: 1 < /', j < n ) in ¿fsuch that 4>ij= [xlVyj], l«.i,y<n, and Ik *.1I <eh.,y¡\ <e' i <i,j <n. Since every weakly continuous functional on J( is also ultraweakly continuous, property (B„) is weaker than (AJ. (Added in proof. It was pointed out by C. Apóstol that (B„ ) and (A „) are in fact equivalent. This fact is not used below.) We recall now from [8] that a linear subspace J(of ££(3^) is said to be reflexive if it contains every operator T g ¿?( Jif ) with the property that Tx G (J(x)~ for every Received by the editors January 31, 1984. 1980 Mathematics Subject Classification. Primary 47C05. ' The author was partially supported by a grant from the National Science Foundation. ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page 139 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use