On isomorphisms of algebras of compactly supported continuous functions

TL;DR

This paper characterizes isomorphisms of algebras of compactly supported continuous functions on locally compact spaces, showing they are essentially induced by space homeomorphisms and field automorphisms, with applications to Fourier analysis.

Contribution

It provides a topological characterization of algebra isomorphisms on function algebras, extending to differentiable functions and groups, and applies to Fourier transform characterization.

Findings

Any algebra isomorphism is a composition of a space homeomorphism and a field automorphism.

In the case of group-preserving maps, the homeomorphism is a group isomorphism.

Application to characterizing Fourier transforms on Schwartz-Bruhat functions.

Abstract

We study the general form of isomorphisms on the algebra of compactly supported complex-valued continuous functions defined on a locally compact Hausdorff space (the proof of which works for the algebra of $C^k-$differentiable functions on a $C^k-$manifold as well). We obtain using only topological techniques, that any such map is a composition of a homeomorphism of the locally compact space (resp. $C^k-$diffeomorphism), and an automorphism of the field of complex numbers. In the particular case when $X$ is a locally compact group, and the map preserves convolution products, the resulting homeomorphism is also a group isomorphism. An application of this gives a characterisation of the Fourier transform on the algebra of Schwartz-Bruhat functions on locally compact Abelian groups.