Vol. 35, No. 3, 1970

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Principal ideals in F-algebras

Ronn L. Carpenter

Vol. 35 (1970), No. 3, 559–563

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.

Mathematical Subject Classification
Primary: 46.55
Secondary: 32.00
Received: 11 September 1969
Published: 1 December 1970
Ronn L. Carpenter