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 operates the algebra is C(Δ). 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.
operates the algebra is 