Isomorphisms of AC(σ) spaces

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.