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

TL;DR

This paper characterizes when certain subspaces of bounded, countably supported functions embed into , showing a key condition involving the absence of isometric copies of c_0() and constructing a specific counterexample with an unconditional basis.

Contribution

It provides a characterization of Banach subspaces of  with countable support that embed into , and constructs a subspace with an unconditional basis that does not embed into  and has separable weakly compact subsets.

Findings

Subspaces of  embed into  iff they do not contain isometric copies of c_0().

Constructed a subspace with an unconditional basis that does not embed into .

Every weakly compact subset of the constructed subspace is separable.

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 $c_0(\omega_1)$).