Automatic continuity and representation of group homomorphisms defined between groups of continuous functions

Let C ( X , G ) be the group of continuous functions from a topological space X into a topological group G with pointwise multiplication as the composition law, endowed with the uniform convergence topology. To what extent does the group structure of C ( X , G ) determine the topology of X? More generally, when does the existence of a group homomorphism H between the groups C ( X , G ) and C ( Y , G ) implies that there is a continuous map h of Y into X such that H is canonically represented by h? We prove that, for any topological group G and compact spaces X and Y, every non-vanishing C-isomorphism (defined below) H of C ( X , G ) into C ( Y , G ) is automatically continuous and can be canonically represented by a continuous map h of Y into X. Some applications to specific groups and examples are given in the paper.