Local Power Series Quotients of Commutative Banach and Fr echet Algebras

We consider the relationship between derivations and local power series quotients for a locally multiplicatively convex Fr echet algebra (this includes the case of a Banach algebra). In x2 we derive necessary conditions for a commutative Fr echet algebra to have a local power series quotient. Our main result here is Proposition 2.6, which shows that if the generating element has nite closed descent, the algebra cannot be simply a radical algebra with identity adjoined { it must have non-trivial representation theory; if the generating element does not have nite 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 x3 we impose some additional conditions which are automatic for a Banach algebra but required in the case of a Fr echet algebra in order to use standard techniques from representation theory. We consider with which strictly irreducible representations the discontinuity of a derivation must be associated. The main result in this section is Proposition 3.14, which shows that when consideration is xed upon a single seminorm, the exceptional set of irreducible representations supporting the discontinuity must be a nite set. We also prove that derivations on commutative Fr echet algebras whose structure spaces are compact metric in the weak* topology have only nitely many such exceptional points overall. This leads naturally to the case in x4 where we consider a derivation D on a commutative radical Fr echet algebra R with identity adjoined. We show in Theorem 4.8 that a derivation D whose discontinuity is not concentrated in the (Jacobson) radical forces R to have a local power series quotient. The question whether such a derivation can have a separating ideal so large it actually contains the identity element has been recently settled in the aÆrmative by C. J. Read. y The author thanks Pomona College for support as a Visiting Scholar during the summer of the Banach Algebras 1999 conference and the Centre for Mathematics and its Applications for support during the Banach Spaces, Operators, and Algebras Symposium in January 2001 at the Australian National University