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

Piotr Drygier ¹   ; Grzegorz Plebanek ¹

¹ Instytut Matematyczny Uniwersytet Wrocławski Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Fundamenta Mathematicae, Tome 237 (2017) no. 2, pp. 165-186

Résumé

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.

Détail

DOI : 10.4064/fm263-6-2016

Keywords: investigate which compactifications gamma omega discrete space natural numbers omega natural copy banach space complemented gamma omega particular separability remainder gamma omega setminus omega neither sufficient nor necessary complemented gamma omega latter result proved under continuum hypothesis analyse context compactifications omega related embeddings measure algebra omega mathop fin nolimits prove banach space contains rich family complemented copies whenever compact space admits only measures countable maharam type