Real compactness via real maximal ideals of $B_1(X)$

In this paper, constructing a class of ideals of $B_1(X)$ from proper ideals of $C(X)$ a one-one correspondence between the class of real maximal ideals of $C(X)$ and those of $B_1(X)$ is established. The collection of all real maximal ideals of $B_1(X)$ with hull-kernel topology is proved to be homeomorphic to the space of real maximal ideals of $C(X)$ endowed with a topology finer than the subspace topology induced from its structure space. It is also proved that a Tychonoff space is real compact if and only if every real maximal ideal of $B_1(X)$ is fixed. As a consequence, within the class of real compact $T_4$ spaces whose points are $G_\delta$, $B_1(X) = {B_1}^*(X)$ if and only if $X$ is finite.