Local power series quotients of commutative Banach and Fréchet algebras

Abstract:

We consider the relationship between derivations and local power series quotients for a locally multiplicatively convex Fréchet algebra. In §2 we derive necessary conditions for a commutative Fréchet algebra to have a local power series quotient. Our main result here is Proposition 2.6, which shows that if the generating element has finite closed descent, the algebra cannot be simply a radical algebra with identity adjoined—it must have nontrivial representation theory; if the generating element does not have finite closed descent, then the algebra cannot be a Banach algebra, and the generating element must be locally nilpotent (but non-nilpotent) in an associated quotient algebra. In §3 we consider a fundamental situation which leads to local power series quotients. Let $D$ be a derivation on a commutative radical Fréchet algebra ${\mathcal {R}}^{\sharp }$ with identity adjoined. We show in Theorem 3.10 that if the discontinuity of $D$ is not concentrated in the (Jacobson) radical, then ${\mathcal {R}}^{\sharp }$ has a local power series quotient. The question of whether such a derivation can have a separating ideal so large it actually contains the identity element has been recently settled in the affirmative by C. J. Read.