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, their algebraic tensor product, and the completion of with respect to a uniform crossnorm (where is the ``least", and the greatest, crossnorm). In §2 we characterize the relatively compact subsets of as those which, considered as spaces of operators from to *F* and from to *E*, take the unit balls in and in to relatively compact sets in *F* and *E*, respectively. In §3 we prove that if and are compact operators then and are each compact, and results concerning the problem for an arbitrary crossnorm are also given. Schatten has characterized as a certain space of operators of ``finite -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:
https://doi.org/10.1090/S0002-9939-1972-0326458-7

Keywords:
Tensor product,
space of operators,
compact operator,
weak operator topology

Article copyright:
© Copyright 1972
American Mathematical Society