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

An approximation property related to $ M$-ideals of compact operators


Authors: Rafael Payá and Wend Werner
Journal: Proc. Amer. Math. Soc. 111 (1991), 993-1001
MSC: Primary 46B20; Secondary 47B07, 47D15, 47D30
MathSciNet review: 1039261
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Abstract: We investigate a variant of the compact metric approximation property which, for subspaces $ X$ of $ {c_0}$, is known to be equivalent to $ K(X)$, the space of compact operators on $ X$, being an $ M$-ideal in the space of bounded operators on $ X,L(X)$. Among other things, it is shown that an arbitrary Banach space $ X$ has this property iff $ K(Y,X)$ is an $ M$-ideal in $ L(Y,X)$ for all Banach spaces $ Y$ and, furthermore, that $ X$ must contain a copy of $ {c_0}$. The proof of the central theorem of this note uses a characterization of those Banach spaces $ X$ for which $ K(X)$ is an $ M$-ideal in $ L(X)$ obtained earlier by the second author, as well as some techniques from Banach algebra theory.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1039261-1
PII: S 0002-9939(1991)1039261-1
Article copyright: © Copyright 1991 American Mathematical Society