Approximate local isometries on spaces of absolutely continuous functions

Abstract

Let AC(X) be the Banach algebra of all absolutely continuous complex-valued functions f on a compact subset X ⊂ R with at least two points under the norm ||f||_Σ=||f||_∞+V(f), where V(f) denotes the total variation of f. We prove that every approximate local isometry from AC(X) to 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 AC(X) onto 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 AC(X), but they hold for the sets of isometric reflections and generalized bi-circular projections of AC(X).