On Structure space of the ring $B_1(X)$

Abstract

In this article, we continue our study of the ring of Baire one functions on a topological space $(X,\tau)$, 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 $(X,\tau)$ 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 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 $\sigma$ than the original $T_4$ topology $\tau$ on $X$, it is proved that $B_1(X)$ contains free (maximal) ideals if $\sigma$ is strictly finer than $\tau$. It is also proved that $\tau = \sigma$ if and only if $B_1(X) = C(X)$. Moreover, in the class of all perfectly normal $T_1$ spaces, $B_1(X) = C(X)$ is equivalent to the discreteness of the space $X$.