This is an announcement for the paper “Isomorphisms of $AC(\sigma)$ spaces for countable sets” by Ian Dousthttps://arxiv.org/find/math/1/au:+Doust_I/0/1/0/all/0/1, Shaymaa Al-shakarchihttps://arxiv.org/find/math/1/au:+Al_shakarchi_S/0/1/0/all/0/1.
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.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1710.09073