Isomorphisms of $BV(\sigma)$ spaces

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.