Isomorphisms of $AC(\sigma )$ spaces

Ian Doust ¹   ; Michael Leinert ²

¹ School of Mathematics and Statistics University of New South Wales Sydney 2052 Australia
² Institut für Angewandte Mathematik Universität Heidelberg Im Neuenheimer Feld 294 D-69120 Heidelberg, Germany

Studia Mathematica, Tome 228 (2015) no. 1, pp. 7-31

Résumé

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.

Détail

DOI : 10.4064/sm228-1-3

Keywords: analogues classical banach stone theorem spaces continuous functions studied context spaces absolutely continuous functions introduced ashton doust sigma algebra isomorphic sigma sigma homeomorphic sigma converse however false positive direction converse implication does sets sigma sigma confined restricted collection compact sets set simple polygons