Approximate Local Isometries on Spaces of Absolutely Continuous Functions

Abstract

Let $$\mathrm {AC}(X)$$ be the Banach algebra of all absolutely continuous complex-valued functions f on a compact subset $$X\subset \mathbb {R}$$ with at least two points under the norm $$\left\| f\right\| _{\Sigma }=\left\| f\right\| _\infty +\mathrm {V}(f)$$ , where $$\mathrm {V}(f)$$ denotes the total variation of f. We prove that every approximate local isometry from $$\mathrm {AC}(X)$$ to $$\mathrm {AC}(Y)$$ admits a Banach–Stone type representation as an isometric weighted composition operator. Using this description, we prove that the set of linear isometries from $$\mathrm {AC}(X)$$ onto $$\mathrm {AC}(Y)$$ is algebraically reflexive and 2-algebraically reflexive. Moreover, it is shown that although the topological reflexivity and 2-topological reflexivity do not necessarily hold for the isometry group of $$\mathrm {AC}(X)$$ , but they hold for the sets of isometric reflections and generalized bi-circular projections of $$\mathrm {AC}(X)$$ .