## $M$-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 $L(\ell _{1},X)$.## References

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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