This is an announcement for the paper “Embedding Banach spaces into the space of bounded functions with countable support” by William B. Johnsonhttps://arxiv.org/search/math?searchtype=author&query=Johnson%2C+W+B, Tomasz Kaniahttps://arxiv.org/search/math?searchtype=author&query=Kania%2C+T.
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 $c_0(\omega_1)$).
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.05239