A general Banach—Stone type theorem and applications

Luiz Gustavo Cordeiro (luiz.cordeirorand@ufsc.br)
Previously at Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon
Currently at Departamento de Matemática, Universidade Federal de Santa Catarina

The author was supported by CAPES/Ciência Sem Fronteiras PhD scholarship 012035/2013-00 and by ANR project GAMME (ANR-14-CE25-0004)
Please refer to the originally published version:
DOI: 10.1016/j.jpaa.2019.106275

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.