Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces

Abstract

We study $M$-ideals of compact operators by means of the property~$(M)$ introduced in \cite{Kal-M}. Our main result states for a separable Banach space $X$ that the space of compact operators on $X$ is an $M$-ideal in the space of bounded operators if (and only if) $X$ does not contain a copy of ${\ell }_1$, has the metric compact approximation property, and has property~$(M)$. The investigation of special versions of property~$(M)$ leads to results on almost isometric structure of some classes of Banach spaces. For instance, we give a simple necessary and sufficient condition for a Banach space to embed almost isometrically into an ${\ell }_p$-sum of finite-dimensional spaces resp.\ into $c_0$, and for $2<p<\iy$ we prove that a subspace of $L_p$ embeds almost isometrically into ${\ell }_p$ if and only if it does not contain a subspace isomorphic to ${\ell }_2$.

References in corpus (1)

• The $M$-ideal structure of some algebras of bounded linear operators