This is an announcement for the paper “Caompactifications of $\omega$ and the Banach Space $c_0$ by Piotr Drygier, Grzegorz Plebanek.

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. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1601.03770