Isomorphisms of $AC(\sigma )$ spaces for countable sets

Abstract

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

14 pages. Revised version with slightly expanded introduction. Some minor typos corrected. (To appear in the Proceedings of the 28th IWOTA, in "Operator Theory: Advances and Applications".)

Cited by in corpus (4)

• $AC(\sigma )$ spaces for polygonally inscribed curves
• The Banach algebras $AC(\sigma )$ and $BV(\sigma )$
• Isomorphisms of $BV(\sigma )$ spaces
• Isomorphisms of $AC(\sigma )$ spaces for linear graphs