$AC(\sigma)$ spaces for polygonally inscribed curves

TL;DR

This paper characterizes the algebra of absolutely continuous functions on certain polygonally inscribed curves, showing it is uniquely determined by the set's homeomorphism class, extending classical function space theorems.

Contribution

It introduces a new class of compact sets made of convex curves and proves they have the Gelfand--Kolmogorov property for their function algebras.

Findings

The algebra of absolutely continuous functions is determined by the set's topology.

Finite unions of convex curves form a class with the Gelfand--Kolmogorov property.

The results extend classical theorems to new geometric settings.

Abstract

For certain families of compact subsets of the plane, the isomorphism class of the algebra of absolutely continuous functions on a set is completely determined by the homeomorphism class of the set. This is analogous to the Gelfand--Kolmogorov theorem for $C(K)$ spaces. In this paper we define a family of compact sets comprising finite unions of convex curves and show that this family has the `Gelfand--Kolmogorov' property.