On functional analytic approach for Gleason problem in the theory of SCV

Abstract

This paper establishes the Gleason result for finitely generated ideals in the context of Frechet algebras, and, in particular, provides an affirmative answer to a question about the Gleason result in commutative Frechet algebras (Carpenter posed this question for uniform Frechet algebras in 1970). As a welcome bonus of our method, locally Stein algebras are completely characterized, and, as an application of this characterization, an affirmative answer to the Gleason problem (1964) for such algebras is provided recapturing all the classical results on the Gleason problem in the theory of several complex variables.