Isometries on Banach algebras of vector-valued maps

TL;DR

This paper develops a unified framework to analyze isometries on Banach algebras of vector-valued Lipschitz and differentiable maps, revealing their canonical forms and extending previous results.

Contribution

It introduces the concept of admissible quadruples to characterize isometries on vector-valued function algebras, generalizing prior work and confirming known conjectures.

Findings

Isometries on these algebras have a canonical form

The approach applies to maps taking values in unital commutative C*-algebras

The framework unifies the study of Lipschitz and differentiable map algebras

Abstract

We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of admissible quadruples. We describe isometries on function spaces of some admissible quadruples that take values in unital commutative $C^*$-algebras. As a consequence we confirm the statement of \cite[Example 8]{jp} on Lipschitz algebras and show that isometries on such algebras indeed take the canonical form.