Uniqueness of the Fréchet algebra topology on certain Fréchet algebras
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.