Realcompactness and Banach-Stone theorems

TL;DR

This paper characterizes linear biseparating maps and isometries between vector-valued continuous function spaces on realcompact spaces, extending classical Banach-Stone theorems.

Contribution

It provides a complete description of linear biseparating maps and isometries for vector-valued continuous function spaces on realcompact spaces, with focus on bounded functions.

Findings

Characterization of linear biseparating maps between vector-valued continuous function spaces.

Description of linear isometries between bounded and uniformly continuous vector-valued function spaces.

Application of results to classical Banach-Stone theorems.

Abstract

For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.