Uniqueness of the Frechet algebra topology on certain Frechet algebras

2014/10/07 by Shital R. Patel, Patel, Shital R.
Mathematics · #46H40 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 46J05 #Secondary 13F25 #math.FA #msc:13F25 #msc:46H40 #msc:46J05

paper · pdf · doi:10.48550/arxiv.1410.1695

40 pages, affirmative solution of an unsolved problem (1978) in the theory of automatic continuity

arxiv created 2014/10/07 · arxiv updated 2014/10/08

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.

Related

• On affirmative solution to Michael's acclaimed problem in the theory of Fréchet algebras, with applications to automatic continuity theory
• On discontinuity of derivations, inducing non-unique complete metric topologies
• Topology and geometry of the general composition of formal power series -- towards Fréchet-Lie group-like formalism
• Frechet algebras in abstract Harmonic analysis
• Non-Archimedean Fréchet Algebras and the Loop Space of a Hypersurface Complement