Isomorphisms of AC(σ) spaces for countable sets

Abstract

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

The Banach algebras $AC(σ)$ and $BV(σ)$

“…A number of gaps in some of the earlier arguments were also identified. There is now much more known about the structure of these spaces (see [1,2,16,17]) and new applications have arisen (see, for example, [21]). Unfortunately this provides a challenge for readers, as foundational results are somewhat scattered through the literature and were often shown using different (but equivalent) definitions of variation.…”