Hyperreflexivity and a Dual Product Construction1

We show that an example of a nonhyperreflexive CSL algebra recently constructed by Davidson and Power is a special case of a general and natural reflexive subspace construction. Completely different techniques of proof are needed because of absence of symmetry. It is proven that if y and ¿T are reflexive proper linear subspaces of operators acting on a separable Hubert space, then the hyperreflexivitv constant of (5"x ®3~\ ) -1 is at least as great as the product of the constants of y and ST. This paper was inspired by the interesting " key example" in the recent paper [2] by Davidson and Power in which a nonhyperreflexive CSL algebra was constructed. In an attempt to completely understand this result we obtained a distance constant inequality of a more general nature, which we present here. Let H, K be separable Hubert spaces—finite or infinite dimensional—and let y, ST be linear subspaces of £(//), L(K), which are reflexive in the LoginovShulman sense, (if is reflexive iff whenever T G L(H) is such that Tx G [Sx], x g H, then F g y, where [•] means closure.) Let c€(Sf), Jf(¿T) be the constants of hyperreflexivity of if and 3~ as defined in [4]. We recall that a subspace if of £(//) is hyperreflexive if there is a constant C such that for operators T in £(//), i/(F,y)< Csup{||F±Fö||: P, Q are projections with P^ifQ = 0}. and the optimal constant is denoted ff(if ). If if is reflexive but not hyperreflexive, then we define 3i~(if ) = + oo. We make use of preannihilator techniques, and refer the reader to [1, 4, 5, 7] for details. As shown in [4], the reflexive subspaces of L(H) are precisely those for which the preannihilator in £P* = Cx is the || ■ ||,-closed linear span of rank < 1 operators, where || • ||, denotes trace-class norm. Since ¿f± , ¿F± are generated by rank < 1 operators, so is the tensor product of preannihilators if^ ®y± . By this we mean the || • ||,-closed linear subspace of the ideal of trace-class operators on £(// ® AT) generated by the elementary tensors {/®g: f^ifx, g g Sf±), where H <8> AT denotes the usual tensor product Hilbert space. (When we write if®3~, we will mean the cr-weakly closed linear subspace of £(// ® A") generated by {S ® T: S g if, T g y}.) Thus the annihilator (<fx ®f±)x = [A G L(H ® K):Tr(Ah) = 0, h g if± ®T±) Received by the editors April 15, 1985. 1980 Mathematics Subject Classification. Primary 47D25; Secondary 47A15. 46L10.