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(σ₁) is algebra isomorphic to AC(σ₂) then σ₁ is homeomorphic to σ₂. The converse however is false. In a positive direction we show that the converse implication does hold if the sets σ₁ and σ₂ are confined to a restricted collection of compact sets, such as the set of all simple polygons.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm228-1-3,
author = {Ian Doust and Michael Leinert},
title = {Isomorphisms of AC($\sigma$) spaces},
journal = {Studia Mathematica},
volume = {231},
year = {2015},
pages = {7-31},
zbl = {06497977},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm228-1-3}
}
Ian Doust; Michael Leinert. Isomorphisms of AC(σ) spaces. Studia Mathematica, Tome 231 (2015) pp. 7-31. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm228-1-3/