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

TL;DR

This paper advances the understanding of the Gleason problem in the context of Frechet algebras, providing new results and characterizations that connect functional analysis with complex variables.

Contribution

It establishes the Gleason result for finitely generated ideals in Frechet algebras and characterizes locally Stein algebras, answering longstanding questions.

Findings

Gleason result proven for finitely generated ideals in Frechet algebras

Locally Stein algebras characterized completely

Affirmative solution to the Gleason problem for locally Stein algebras

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.