Isomorphisms of $AC(\sigma )$ spaces
Volume 228 / 2015
Studia Mathematica 228 (2015), 7-31 MSC: Primary 46J10; Secondary 46J35, 47B40, 26B30. DOI: 10.4064/sm228-1-3
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.