Detaching Maps Between Spaces of Continuous Functions

Abstract

Let C(S) and C(T) denote the spaces of real or complex-valued con- tinuous functions on the Tihonov spaces S and T,respectively. An additive operator H:C(T) -C(S)isseparatingif,forI,r€C(T),Iz=0÷ HxHx=0. In[3]it isshownthatifHisabiseparatingmap(bothHandH-1are separating then the realcompactifications ofS and T are homeomorphic. If H is linear and S and T are realcompact then H is continuous [4]. We investigate weaker conditions on a separating map H than biseparating which imply that H is continuous. For instance, it is shown in theorem 4.2 that if S and UT are locally compact, S connected, H injective and "detaching", then H is a "weighted homomorphism"; such a map is continuous ifT is realcompact.

Keywords

• separating map
• detaching map
• rings of continuous functions