Vol. 16, No. 1, 1966

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Commutative F-algebras

Melvin Rosenfeld

Vol. 16 (1966), No. 1, 159–166

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.

Mathematical Subject Classification
Primary: 46.55
Received: 8 September 1964
Published: 1 January 1966
Melvin Rosenfeld