Paper detail

On isomorphisms of algebras of compactly supported continuous functions

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.