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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Compactness in topological tensor products and operator spaces


Author: J. R. Holub
Journal: Proc. Amer. Math. Soc. 36 (1972), 398-406
MSC: Primary 47B05
MathSciNet review: 0326458
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0326458-7
PII: S 0002-9939(1972)0326458-7
Keywords: Tensor product, space of operators, compact operator, weak operator topology
Article copyright: © Copyright 1972 American Mathematical Society