Hyper-Stonean envelopes of compact spaces
Volume 246 / 2019
Studia Mathematica 246 (2019), 31-46
MSC: Primary 46H10, 46J10, 46B25; Secondary 28A33, 28C15.
DOI: 10.4064/sm170815-14-2
Published online: 7 September 2018
Abstract
Let $K$ be a compact space, and denote by $\widetilde{K}$ its hyper-Stonean envelope. We discuss the class of spaces $K$ with the property that $\widetilde{K}$ is homeomorphic to $\widetilde{{\mathbb I}}$, the hyper-Stonean envelope of the closed unit interval ${\mathbb I}$. Certainly each uncountable, compact, metrizable space $K$ belongs to this class. We describe several further classes of compact spaces $K$ for which $\widetilde{K} = \widetilde{{\mathbb I}}$. In fact, $\widetilde{K} = \widetilde{{\mathbb I}}$ if and only if the Banach spaces $M(K)$ and $M({\mathbb I})$ of measures on $K$ and ${\mathbb I}$ are isometrically isomorphic.
