Isometries and approximate local isometries between AC^p(X)-spaces

Abstract

Let X and Y be compact subsets of R with at least two points. For p ≥ 1, let AC^p(X) be the space of all absolutely continuous complex-valued functions f on X such that f' in L^p(X), with the sum norm. e describe the topological reflexive closure of the set of linear isometries from AC^p(X) onto AC^p(Y ). Using this description, we prove that such a set is algebraically reflexive and 2-algebraically reflexive. Moreover, as another application, it is shown that the sets of isometric reflections and generalized bi-circular projections of AC^p(X) are topologically reflexive and 2-topologically reflexive.