Commutative<i>F</i>-algebras

Abstract

We extend several theorems for commutative Banach algebras to topological algebras with a sequence of semi-norms (F-algebras). The question of what functions "operate" on an F-algebra is considered. It is proven that analytic functions in several complex variables operate by applying a theorem due to Waelbroeck. If all continuous functions operate on an F-algebra, then it is an algebra of continuous functions. However, unlike the situation for Banach algebras [6], it is not true that if V operates the algebra is C(). This will be shown by an example. A theorem due to Curtis [4], concerning continuity of derivations when the algebra is regular is extended to F-algebras. The result is applied to an algebra of Lipschitz functions to show that it has only a trivial derivation.