Uniqueness of the Frechet algebra topology on certain Frechet algebras

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.