Uniqueness of the Frechet algebra topology on certain Frechet algebras

TL;DR

This paper proves the uniqueness of the Frechet algebra topology for certain power series algebras, resolving a question posed in 1978 and exploring implications for automatic continuity.

Contribution

It characterizes Frechet algebras of power series and confirms the uniqueness of their topology, extending results to infinitely many indeterminates.

Findings

Frechet algebra topology is unique for power series algebras in k indeterminates.

Beurling-Frechet algebras of semiweight type do not satisfy Loy's equicontinuity condition.

Applications to automatic continuity in infinite-dimensional cases.

Abstract

In 1978, Dales posed a question about the uniqueness of the (F)-algebra topology for (F)-algebras of power series in k indeterminates. We settle this in the affirmative for Frechet algebras of power series in k indeterminates. The proof goes via first completely characterizing these algebras; in particular, it is shown that the Beurling-Frechet algebras of semiweight type do not satisfy a certain equicontinuity condition due to Loy. Some applications to the theory of automatic continuity are also given, in particular the case of Frechet algebras of power series in infinitely many indeterminates.