On structure space of the ring B1(X)

Abstract

In this article, we continue our study of the ring of Baire one functions on a topological space (X,?), denoted by B1(X), and extend the well known M. H. Stones?s theorem from C(X) to B1(X). Introducing the structure space of B1(X), an analogue of Gelfand-Kolmogoroff theorem is established. It is observed that (X,?) may not be embedded inside the structure space of B1(X). This observation inspired us to introduce a weaker form of embedding and show that in case X is a T4 space, X is weakly embedded as a dense subspace, in the structure space of B1(X). It is further established that the ring B*1(X) of all bounded Baire one functions, under suitable conditions, is a C-type ring and also, the structure space of B*1(X) is homeomorphic to the structure space of B1(X). Introducing a finer topology ? than the original T4 topology ? on X, it is proved that B1(X) contains free maximal ideals if ? is strictly finer than ?. Moreover, in the class of all perfectly normal T1 spaces, ? = ? is necessary as well as sufficient for B1(X) = C(X).