Maximal ideals in subalgebras of $C(X)$

Abstract:

Let $X$ be a completely regular space, and let $A(X)$ be a subalgebra of $C(X)$ containing ${C^ * }(X)$. We study the maximal ideals in $A(X)$ by associating a filter $Z(f)$ to each $f \in A(X)$. This association extends to a one-to-one correspondence between $\mathcal {M}(A)$ (the set of maximal ideals of $A(X)$) and $\beta X$. We use the filters $Z(f)$ to characterize the maximal ideals and to describe the intersection of the free maximal ideals in $A(X)$. Finally, we outline some of the applications of our results to compactifications between $\upsilon X$ and $\beta X$.