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Isomorphisms of $AC(\sigma )$ spaces

Volume 228 / 2015

Ian Doust, Michael Leinert Studia Mathematica 228 (2015), 7-31 MSC: Primary 46J10; Secondary 46J35, 47B40, 26B30. DOI: 10.4064/sm228-1-3

Abstract

Analogues of the classical Banach–Stone theorem for spaces of continuous functions are studied in the context of the spaces of absolutely continuous functions introduced by Ashton and Doust. We show that if $AC(\sigma _1)$ is algebra isomorphic to $AC(\sigma _2)$ then $\sigma _1$ is homeomorphic to $\sigma _2$. The converse however is false. In a positive direction we show that the converse implication does hold if the sets $\sigma _1$ and $\sigma _2$ are confined to a restricted collection of compact sets, such as the set of all simple polygons.

Authors

  • Ian DoustSchool of Mathematics and Statistics
    University of New South Wales
    Sydney 2052 Australia
    e-mail
  • Michael LeinertInstitut für Angewandte Mathematik
    Universität Heidelberg
    Im Neuenheimer Feld 294
    D-69120 Heidelberg, Germany
    e-mail

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