Picard–Borel ideals in Fréchet algebras and Michael's problem

Abstract

A Picard–Borel algebra is a commutative unital complex algebra A $A$ such that every family of pairwise linearly independent invertible elements of A $A$ is linearly independent, and a Picard–Borel ideal I $I$ in a commutative complex unital algebra A $A$ is an ideal of I $I$ of A $A$ such that the quotient algebra A / I $A/I$ is a Picard–Borel algebra. In a preliminary paper, the author proved that every commutative unital Fréchet algebra which is a Picard–Borel algebra is an integral domain. The main result of the present paper is that all Picard–Borel ideals in commutative unital Fréchet algebras are prime. This result seems to be relevant for Michael's problem, since dense Picard–Borel ideals could play a role in the construction of discontinuous characters on some commutative unital Fréchet algebras.