  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.
