On Structure Space of the Ring B1(X)

In this article, we continue our study of the ring of Baire one functions on a topological space (X, tau), 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, tau) 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 sigma than the original T4 topology tau on X, it is proved that B1(X) contains free maximal ideals if sigma is strictly finer than tau. Moreover, in the class of all perfectly normal T1 spaces, sigma = tau is necessary as well as sufficient for B1(X) = C(X).