Suppose B is a commutative
Banach algebra with unit. Gleason has proved that if I is a finitely generated
maximal ideal in B, then there is an open neighborhood U of I in the spectrum of B
such that U is homeomorphic in a natural way to an analytic variety and the Gelfand
transforms of elements of B are analytic on this variety. In this paper it is shown that
this result remains valid for principal ideals in uniform F-algebras with locally
compact spectra. From this it follows that if A is an F-algebra of complex valued
continuous functions on its spectrum satisfying (1) the spectrum of A is locally
compact and has no isolated points, and (2) every closed maximal ideal in A is
principal, then the spec . trum of A can be given the structure of a Riemann
surface in such a way that A can be identified with a closed subalgebra
of the algebra of all functions which are analytic on the spectrum of A.
Finally an example is given which shows that neither Gleason’s result nor the
characterization described in the preceding sentence extends to nonuniform
algebras.
