AUTOMATIC CONTINUITY OF ALMOST MULTIPLICATIVE LINEAR FUNCTIONALS ON FRÉCHET ALGEBRAS

Abstract

Abstract. A linear functional T on a Fr´echet algebra (A,(p n )) is calledalmost multiplicative with respect to the sequence (p n ), if there existse≥ 0 such that |Tab− TaTb| ≤ ep n (a)p n (b) for all n∈ N and for everya,b∈ A.We show that an almost multiplicative linear functional on a Fr´echetalgebra is either multiplicative or it is continuous, and hence every almostmultiplicative linear functional on a functionally continuous Fr´echet al-gebra is continuous. 1. IntroductionLet A be an algebra over the complex field. A subset V of A is balanced ifλV ⊆ V for all scalars λ such that |λ| ≤ 1. An algebra A is a Fr´echet algebraif it is a complete metrizable topological linear space and has a neighbourhoodbasis (V n ) of zero such that V n is absolutelyconvex (convex and balanced) andV n is idempotent, i.e., V n V n ⊆ V n for all n ∈ N. The topology of a Fr´echetalgebra A can be generated by a sequence (p n ) of separating submultiplicativeseminorms, i.e., p n (xy) ≤ p n (x)p n (y) for all n ∈ N and x,y ∈ A, such thatp