## Uniqueness of the Fréchet algebra topology on certain Fréchet algebras

### Volume 234 / 2016

Studia Mathematica 234 (2016), 31-47
MSC: Primary 46J05; Secondary 13F25, 46H40.
DOI: 10.4064/sm8132-5-2016
Published online: 20 July 2016

#### 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 Fréchet algebras of power series in $k$ indeterminates. The proof goes via first completely characterizing these algebras; in particular, it is shown that the Beurling–Fréchet 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 to the case of Fréchet algebras of power series in infinitely many indeterminates.