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

TL;DR

This paper establishes a correspondence between the topological structure of linear graph drawings in the plane and the algebraic structure of associated absolutely continuous function spaces, showing they are isomorphic if and only if the graphs are homeomorphic.

Contribution

It provides a novel characterization of linear graph drawings via isomorphisms of their associated $AC(\sigma)$ spaces, extending classical results to this setting.

Findings

Homeomorphic linear graph drawings have isomorphic $AC(\sigma)$ spaces.

Isomorphism of $AC(\sigma)$ spaces characterizes graph homeomorphism.

Analogue of Gelfand-Kolmogorov theorem for linear graph $AC(\sigma)$ spaces.

Abstract

We show that among compact subsets of the plane which are drawings of linear graphs, two sets $\sigma$ and $\tau$ 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 $C(\Omega)$ spaces.