Isomorphisms of $$AC(\sigma )$$ spaces for linear graphs

Abstract

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.