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

TL;DR

This paper explores conditions under which the Banach space c_0 is complemented in spaces of continuous functions on compactifications of natural numbers, revealing that separability of the remainder space is not a determining factor.

Contribution

It establishes that the separability of the remainder of a compactification is neither necessary nor sufficient for c_0 to be complemented in C(γω), and links this to measure algebra embeddings.

Findings

Separability of the remainder space is not sufficient for c_0 complementability.

Under the continuum hypothesis, separability is not necessary for c_0 complementability.

Spaces C(K) with measures of countable Maharam type contain many complemented copies of 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.