Automatic continuity of biseparating homomorphisms defined between groups of continuous functions

Let C ( X , T ) be the group of continuous functions of a compact Hausdorff space X to the unit circle of the complex plane T with the pointwise multiplication as the composition law. We investigate how the structure of C ( X , T ) determines the topology of X. In particular, which group isomorphisms H between the groups C ( X , T ) and C ( Y , T ) imply the existence of a continuous map h of Y into X such that H is canonically represented by h. Among other results, it is proved that C ( X , T ) determines X module a biseparating group isomorphism and, when X is first countable, the automatic continuity and representation as Banach–Stone maps for biseparating group isomorphisms is also obtained.