Journal of Operator Theory

Volume 36, Issue 1, Summer 1996 pp. 21-43.

Preannihilators, the operator approximation property and dual products

Authors: Jon Kraus

$1$ and David R. Larson

$2$
Author institution:

$1$ Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14214, U.S.A.

$2$ Department of Mathematics, Texas A & M University, College Station, TX 77843, U.S.A.

Summary: It was shown by Effros, Kraus and Ruan that an ultraweakly closed subspace of B

$H$ has the weak*-operator approximation property if and only if its predual has the operator approximation property. We show that the preannihilator of such a subspace S has the operator approximation property if and only if any ultraweakly closed subspace T of B

$K$ the dual product of S and T is the ultraweakly closed linear span of the algebraic tensor product of S and B

$K$ and the algebraic tensor product of B

$H$ and T. Using this we also show that there are reflexive subspaces S and T for which the dual product is strictly larger than this ultraweakly closed linear span, answering a question of the second author.

Keywords: Operator space, tensor product, dual product, hyperreflexive, operator approximation property, preannihilator.

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