$M$-ideals of compact operators are separably determined

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 $L(\ell _{1},X)$.