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$M$-ideals of compact operators
are separably determined

Author: Eve Oja
Journal: Proc. Amer. Math. Soc. 126 (1998), 2747-2753
MSC (1991): Primary 46B28, 47D15, 46B20
MathSciNet review: 1469429
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Abstract: We prove that the space $K(X)$ of compact operators on a Banach space $X$ is an $M$-ideal in the space $L(X)$ of bounded operators if and only if $X$ has the metric compact approximation property (MCAP), and $K(Y)$ is an $M$-ideal in $L(Y)$ for all separable subspaces $Y$ of $X$ having the MCAP. It follows that the Kalton-Werner theorem characterizing $M$-ideals of compact operators on separable Banach spaces is also valid for non-separable spaces: for a Banach space $X,\ K(X)$ is an $M$-ideal in $L(X)$ if and only if $X$ has the MCAP, contains no subspace isomorphic to $\ell _{1},$ and has property $(M).$ It also follows that $K(Z,X)$ is an $M$-ideal in $L(Z,X)$ for all Banach spaces $Z$ if and only if $X$ has the MCAP, and $K(\ell _{1},X)$ is an $M$-ideal in $L(\ell _{1},X)$.

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Additional Information

Eve Oja
Affiliation: Institute of Pure Mathematics, Tartu University, Vanemuise 46, EE2400 Tartu, Estonia

Received by editor(s): February 14, 1997
Additional Notes: The author was partially supported by the Estonian Science Foundation Grant 3055.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

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