Compactifications of $\omega$ and the Banach space $c_0$

Abstract

We investigate for which compactifications $\gamma \omega$ of the discrete space of natural numbers $\omega$, the natural copy of the Banach space $c_0$ is complemented in $C(\gamma \omega )$. We show, in particular, that the separability of the remainder of $\gamma \omega$ is neither sufficient nor necessary for $c_0$ being complemented in $C(\gamma \omega )$ (for the latter our result is proved under the continuum hypothesis). We analyse, in this context, compactifications of $\omega$ related to embeddings of the measure algebra into $P(\omega )/fin$. We also prove that a Banach space $C(K)$ contains a rich family of complemented copies of $c_0$ whenever the compact space $K$ admits only measures of countable Maharam type.

20 pages, version of Jan 23, 2016