Topological reflexivity of isometries on algebras of C(Y)-valued Lipschitz maps

Abstract

Let X be a compact metric space and let C(Y) be the space of all complex-valued continuous functions on a Hausdorff compact space Y. We prove that the isometry group of the algebra Lip(X,C(Y)) of all C(Y)-valued Lipschitz maps on X, equipped with the sum norm, is topologically reflexive and 2-topologically reflexive whenever the isometry group of C(Y) is topologically reflexive. The same results are established for the sets of isometric reflections and generalized bi-circular projections of Lip(X).