Isometries on Banach algebras of $C(Y)$-valued maps

TL;DR

This paper introduces a unified framework for studying isometries on Banach algebras of vector-valued Lipschitz and differentiable maps using natural $C(Y)$-valuations, extending previous results.

Contribution

It develops a unified approach to analyze isometries on vector-valued function algebras via natural $C(Y)$-valuations, providing a detailed proof of a key theorem.

Findings

Unified approach to isometries on vector-valued function algebras

Extension of Jarosz's theorem with detailed proof

Framework applicable to Lipschitz and differentiable maps

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 natural $C(Y)$-valuezations that take values in unital commutative $C^*$-algebras. A precise proof of a theorem of Jarosz \cite{ja} is exhibited.