Embedding Banach spaces into the space of bounded functions with countable support

Abstract

We prove that a WLD subspace of the space ${\ell }_{\infty }^c(\Gamma )$ consisting of all bounded, countably supported functions on a set $\Gamma$ embeds isomorphically into ${\ell }_{\infty }$ if and only if it does not contain isometric copies of $c_0({\omega }_1)$. Moreover, a subspace of ${\ell }_{\infty }^c({\omega }_1)$ is constructed that has an unconditional basis, does not embed into ${\ell }_{\infty }$, and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of $c_0({\omega }_1)$).

5 pp