On local isometries between algebras of C(Y)-valued differentiable maps

Abstract

Let K be either the real unit interval [0,1] or the complex unit circle T and let C(Y ) be the space of all complex-valued continuous functions on a compact Hausdorff space Y. We prove that the isometry group of the algebra C^1(K,C(Y )) of all C(Y)-valued continuously differentiable maps on K, equipped with the sum-norm, is topologically reflexive and 2-topologically reflexive whenever the isometry group of C(Y) is topologically reflexive.