It is known that the classical Banach-Stone theorem does not extend to the class of AC(σ) spaces of absolutely continuous functions defined on compact subsets of the complex plane. On the other hand, if σ is restricted to the set of compact polygons, then all the corresponding AC(σ) spaces are isomorphic (as algebras). In this paper we examine the case where σ is the spectrum of a compact operator, and show that in this case one can obtain an infinite family of homeomorphic sets for which the corresponding function spaces are not isomorphic.
Key words AC(σ) spaces, functions of bounded variation, topology of planar graphs MSC (2010) Primary 46J10; Secondary 05C10, 46J45, 47B40, 26B30We show that among compact subsets of the plane which are drawings of linear graphs, two sets σ and τ are homeomorphic if and only if the corresponding spaces of absolutely continuous functions (in the sense of Ashton and Doust) are isomorphic as Banach algebras. This gives an analogue for this class of sets to the well-known result of Gelfand and Kolmogorov for C(Ω) spaces.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.