Automatic continuity of biseparating homomorphisms defined between groups of continuous functions

Abstract

Let C(X,T{double-struck}) be the group of continuous functions of a compact Hausdorff space X to the unit circle of the complex plane T{double-struck} with the pointwise multiplication as the composition law. We investigate how the structure of C(X,T{double-struck}) determines the topology of X. In particular, which group isomorphisms H between the groups C(X,T{double-struck}) and C(Y,T{double-struck}) 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{double-struck}) 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. © 2009 Elsevier B.V.