Isomorphisms of $BV(\sigma)$ spaces

TL;DR

This paper explores the structure of $BV(\sigma)$ spaces related to compact sets in the complex plane, showing that isomorphisms of absolutely continuous function spaces extend to bounded variation spaces, with applications to spectral theory.

Contribution

It proves that isomorphisms between $AC(\sigma)$ spaces extend to $BV(\sigma)$ spaces, linking set topology to function space structure and spectral theory.

Findings

Isomorphisms of $AC(\sigma)$ extend to $BV(\sigma)$ spaces.

Homeomorphic sets yield isomorphic $AC(\sigma)$ spaces.

Application to spectral theory of $AC(\sigma)$ operators.

Abstract

In this paper we investigate the relationship between the properties of a compact set $\sigma \subseteq \mathbb{C}$ and the structure of the space $BV(\sigma)$ of functions of bounded variation (in the sense of Ashton and Doust) defined on $\sigma$. For the subalgebras of absolutely continuous functions on $\sigma$, it is known that for certain classes of compact sets one obtains a Gelfand--Kolmogorov type result: the function spaces $AC(\sigma_1)$ and $AC(\sigma_2)$ are isomorphic if and only if the domain sets $\sigma_1$ and $\sigma_2$ are homeomorphic. Our main theorem is that in this case the isomorphism must extend to an isomorphism of the $BV(\sigma)$ spaces. An application is given to the spectral theory of $AC(\sigma)$ operators.