On a class of subalgebras of $C(X)$ and the intersection of their free maximal ideals

Abstract:

Let $X$ be a Tychonoff space and $A$ a subalgebra of $C(X)$ containing $C^*(X)$. Suppose that $C_K(X)$ is the set of all functions in $C(X)$ with compact support. Kohls has shown that $C_K(X)$ is precisely the intersection of all the free ideals in $C(X)$ or in $C^*(X)$. In this paper we have proved the validity of this result for the algebra $A$. Gillman and Jerison have proved that for a realcompact space $X$, $C_K(X)$ is the intersection of all the free maximal ideals in $C(X)$. In this paper we have proved that this result does not hold for the algebra $A$, in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in $A$. The paper terminates by showing that for any realcompact space $X$, there exists in some sense a minimal algebra $A_m$ for which $X$ becomes $A_m$-compact. This answers a question raised by Redlin and Watson in 1987. But it is still unsettled whether such a minimal algebra exists with respect to set inclusion.