Automatic continuity ofn-homomorphisms between FrÉchet algebras

We impose a condition on a commutative regular Fréchet algebra (A, (pm )) to ensure that A/kerpm is a Fréchet Q-algebra. This implies that if θ is an n-homomorphism on certain Fréchet algebras (A, (pm )) into semisimple commutative Fréchet algebras (B,(qm)) such that θ(kerpm) ⫅ kerqm, for large enough m, then θ is continuous. We also show that if A is a Fréchet Q-algebra, B is a semisimple Fréchet algebra, θ: A → B is a dense range n-homomorphism such that θ(A) is factorizable, and the spectral radius vB is continuous on the separating space (θ), then θ is automatically continuous.