Isomorphisms of $AC(\sigma )$ spaces

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.

arXiv comment25 pages