On structure space of the ring b 1 (x)

Atasi Deb Ray, Atanu Mondal

In this article, we continue our study of the ring of Baire one functions on a topological space (X, τ), denoted by B 1 (X), and extend the well known M. H. Stones's theorem from C(X) to B 1 (X). Introducing the structure space of B 1 (X), an analogue of Gelfand-Kolmogoroff theorem is established. It is observed that (X, τ) may not be embedded inside the structure space of B 1 (X). This observation inspired us to introduce a weaker form of embedding and show that in case X is a T 4 space, X is weakly embedded as a dense subspace, in the structure space of B 1 (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 B 1 (X). Introducing a finer topology σ than the original T 4 topology τ on X, it is proved that B 1 (X) contains free maximal ideals if σ is strictly finer than τ. Moreover, in the class of all perfectly normal T 1 spaces, σ = τ is necessary as well as sufficient for B 1 (X) = C(X)