Compactness in Topological Tensor Products and Operator Spaces

Abstract

Let E and F be Banach spaces, $E \otimes F$ their algebraic tensor product, and $E{ \otimes _\alpha }F$ the completion of $E \otimes F$ with respect to a uniform crossnorm $\alpha \geqq \lambda$ (where $\lambda$ is the “least", and $\gamma$ the greatest, crossnorm). In §2 we characterize the relatively compact subsets of $E{ \otimes _\lambda }F$ as those which, considered as spaces of operators from ${E^ \ast }$ to F and from ${F^ \ast }$ to E, take the unit balls in ${E^ \ast }$ and in ${F^ \ast }$ to relatively compact sets in F and E, respectively. In §3 we prove that if ${T_1}:{E_1} \to {E_2}$ and ${T_2}:{F_1} \to {F_2}$ are compact operators then ${T_1}{ \otimes _\lambda }{T_2}$ and ${T_1}{ \otimes _\lambda }{T_2}$ are each compact, and results concerning the problem for an arbitrary crossnorm $\alpha$ are also given. Schatten has characterized ${(E{ \otimes _\alpha }F)^ \ast }$ as a certain space of operators of “finite $\alpha$-norm". In §4 we show that a space of operators has such a representation if and only if its unit ball is weak operator compact.