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A function is invertible in C(X) if it is never zero, and in C*(X) if
it is bounded away from zero. In an arbitrary A(X), of course, there
is no such description of invertibility which is independent of the
structure of the algebra. Thus in §2 we associate to each noninvertible
f∈A(X) a z-filter
(f ) that is a measure of where
f is ``locally'' invertible in A(X). This correspondence extends to
one between maximal ideals of A(X) and z-ultrafilters on X.
In §3 we use the filters
(f ) to describe the intersection of
the free maximal ideals in any algebra A(X). Finally, our main result
allows us to introduce the notion of A(X)-compactness of which
compactness and realcompactness are special cases. In §4 we show how
the Banach-Stone theorem extends to A(X)-compact spaces.