Isomorphisms of $BV(σ)$ spaces

Abstract

In this paper we investigate the relationship between the properties of a compact set σ ⊆ 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.