Approximate local isometries of Banach algebras of differentiable functions

Abstract

Let X and Y be compact subsets of R such that X and Y coincide with the closures of their interiors. For any n ∈N, let C^{(n)}(X) be the Banach algebra of all n-times continuously differentiable complex-valued functions fon X, with the norm ||f||_C=max_{x∈X}(\sum_{k=0}^n(|f(k)(x)|/k!)). We prove that every approximate local isometry of C^{(n)}(X) to C^{(n)}(Y) is an isometric linear algebra monomorphism multiplied by a fixed n-times continuously differentiable unimodular function. This description allows us to establish the algebraic and 2-algebraic reflexivity of the set of linear isometries of C^{(n)}(X) onto C^{(n)}(Y). Furthermore, this algebraic reflexivity becomes topological whenever X and Y are compact intervals of R. Another application of our main result shows that the sets of isometric reflections and generalized bi-circular projections of C^{(n)}(X) are topologically and 2-topologically reflexive.