FUNCTIONALLY CONTINOUS FRECHET ALGEBRAS

Let (A,(pn)) be a regular Frechet algebra, An denote the completion of A/ker pn with respect to the norm p´n(f+ker pn) = pn(f), and MA denote the character space of A. We first show that if for each f Є A, f lMA= 0 implies that f Є ker pn, then the quatient algebra A/ker pn is a Frechent Q-algebra and bence it is functionally continuous. Using this result we prov that if ø: A → B is surjective homomorphism, where (A,(pn)) is a regular Frechet algebra, (B, qn)) is a semisimple Frechet algebra and such that ø(ker pn) ???? ker qn, for all n, then ø is continuous. Finally, we present certain classes of functionally continuous Frechet algebras.

Keywords:

functionally continuous , regular Frechet algebra , Q-algebra , projective limit , Automatic continuity