Compactifications of $\omega $ and the Banach space $c_0$
Volume 237 / 2017
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 $\gamma \omega \setminus \omega $ is neither sufficient nor necessary for $c_0$ to be complemented in $C(\gamma \omega )$ (the latter 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 )/\mathop {\rm fin}\nolimits $.
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.
