On affirmative solution to Michael's acclaimed problem in the theory of Fr\'{e}chet algebras, with applications to automatic continuity theory

TL;DR

This paper affirms Michael's longstanding problem on the functional continuity of commutative Fréchet algebras, extending the result to non-commutative cases and introducing new approaches and applications in automatic continuity theory.

Contribution

It provides the first affirmative solution to Michael's problem for all Fréchet algebras, including non-commutative cases, and develops two novel approaches to attack these problems.

Findings

Confirmed the automatic continuity of all Fréchet algebras.

Developed two approaches: one using formal power series, another generating alternative topologies.

Extended results to non-commutative Fréchet algebras.

Abstract

In 1952, Michael posed a question about the functional continuity of commutative Frechet algebras in his memoir, known as Michael problem in the literature. We settle this in the affirmative along with its various equivalent forms, even for the non-commutative case. Indeed, we continue our recent works, and develop two approaches to directly attack these problems. The first approach is to show that the test case for this problem, the Frechet algebra of all entire functions on the Banach space of all bounded complex sequences, is, in fact, a Frechet algebra of all complex formal power series in one indeterminate, if there exists a discontinuous character. In the second approach, the existence of a discontinuous character would allow us to generate other Frechet algebra topology, inequivalent to the usual Frechet algebra topology, by applying the method of Read (he used this method to show that the famous Singer-Wermer conjecture (1955) fails in the Frechet case). In both the approaches, an important tool is a topological version of the (symmetric) tensor algebra over a Banach space; the elementary, but crucial, idea is to express the test algebra as a weighted Frechet symmetric algebra over the Banach space of all absolutely summable complex sequences. Several mathematicians have worked on two problems of Michael since 1952, giving affirmative solutions for special classes of Frechet algebras under various conditions, or discussing various test cases, or discussing various approaches, or discussing various other equivalent forms, or deriving other important automatic continuity results such as the (non-)uniqueness of the Frechet algebra topology for certain commutative Frechet algebras by alternate, difficult or lengthy methods. We summarize effects of our affirmative solutions on these attempts in addition to giving various new (important) applications in automatic continuity theory.