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 $\mathrm {AC}(X)$.