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We study the general form of isomorphisms on the algebra of compactlysupported complex-valued continuous functions defined on a locally compactHausdorff space (the proof of which works for the algebra of$C^k-$differentiable functions on a $C^k-$manifold as well). We obtain usingonly topological techniques, that any such map is a composition of ahomeomorphism of the locally compact space (resp. $C^k-$diffeomorphism), and anautomorphism of the field of complex numbers. In the particular case when $X$is a locally compact group, and the map preserves convolution products, theresulting homeomorphism is also a group isomorphism. An application of thisgives a characterisation of the Fourier transform on the algebra ofSchwartz-Bruhat functions on locally compact Abelian groups.

On isomorphisms of algebras of compactly supported continuous functions