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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$M$-ideals of compact operators are separably determined
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by Eve Oja PDF
Proc. Amer. Math. Soc. 126 (1998), 2747-2753 Request permission

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
  • Email: eveoja@math.ut.ee
  • 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
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 2747-2753
  • MSC (1991): Primary 46B28, 47D15, 46B20
  • DOI: https://doi.org/10.1090/S0002-9939-98-04600-0
  • MathSciNet review: 1469429