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Hyper-Stonean envelopes of compact spaces

Volume 246 / 2019

H. Garth Dales, Grzegorz Plebanek 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.

Authors

  • H. Garth DalesDepartment of Mathematics and Statistics
    University of Lancaster
    Lancaster LA1 4YF, United Kingdom
    e-mail
  • Grzegorz PlebanekInstitute of Mathematics
    Wrocław University
    50-384 Wrocław, Poland
    e-mail

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