A general Banach-Stone type theorem and applications

TL;DR

This paper introduces a comprehensive framework unifying various Banach-Stone type theorems, enabling new applications in groupoid algebras and circle-valued function groups through the analysis of disjointness relations.

Contribution

It presents a general framework that unifies existing Banach-Stone theorems and extends their applicability to new mathematical structures.

Findings

Unified framework for Banach-Stone theorems

Application to groupoid algebras

Analysis of disjointness relations

Abstract

One important class of tools in the study of the connections between algebraic and topological structures are the "Banach-Stone type theorems", which describe algebraic isomorphisms of algebras (or groups, lattices, etc.) of functions in terms of homeomorphisms between the underlying topological spaces. Several such theorems have been proven throughout the last century, however not all of them are comparable, and in particular no single one is the strongest. In this article, we describe a general framework which encompasses several of these results, and which allows for new applications related to groupoid algebras, and to groups of circle-valued functions. This is attained by a detailed study of "disjointness relations" on sets of functions, which play a central role (even if not explicitly) in previously-proven Banach-Stone type theorems.