Isomorphisms of $AC(\sigma)$ spaces for countable sets

TL;DR

This paper investigates the isomorphism classes of $AC(\sigma)$ spaces for spectra of compact operators, revealing that unlike polygonal spectra, these spaces can be non-isomorphic despite topological similarities.

Contribution

It extends understanding of $AC(\sigma)$ spaces by showing spectra of compact operators can produce non-isomorphic function spaces, contrasting with the polygon case.

Findings

$AC(\sigma)$ spaces for polygons are all isomorphic

Spectra of compact operators can yield non-isomorphic $AC(\sigma)$ spaces

The classical Banach--Stone theorem does not fully extend to $AC(\sigma)$ spaces

Abstract

It is known that the classical Banach--Stone theorem does not extend to the class of $AC(\sigma)$ spaces of absolutely continuous functions defined on compact subsets of the complex plane. On the other hand, if $\sigma$ is restricted to the set of compact polygons, then all the corresponding $AC(\sigma)$ spaces are isomorphic. In this paper we examine the case where $\sigma$ 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.