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Isomorphisms of $BV(σ)$ spaces

In this paper we investigate the relationship between the properties of a compact set $σ\subseteq \mathbb{C}$ and the structure of the space $BV(σ)$ of functions of bounded variation (in the sense of Ashton and Doust) defined on $σ$. For the subalgebras of absolutely continuous functions on $σ$, it is known that for certain classes of compact sets one obtains a Gelfand--Kolmogorov type result: the function spaces $AC(σ_1)$ and $AC(σ_2)$ are isomorphic if and only if the domain sets $σ_1$ and $σ_2$ are homeomorphic. Our main theorem is that in this case the isomorphism must extend to an isomorphism of the $BV(σ)$ spaces. An application is given to the spectral theory of $AC(σ)$ operators.