A class of factorable topological algebras

The famous Cohen factorization theorem, which says that every Banach algebra with bounded approximate identity factors, has already been generalized to locally convex algebras with what may be termed “uniformly bounded approximate identities”. Here we introduce a new notion, that of fundamentality generalizing both local boundedness and local convexity, and we show that a fundamental Fréchet algebra with uniformly bounded approximate identity factors. Fundamentality is a topological vector space property rather than an algebra property. We exhibit some non-fundamental topological vector space and give a necessary condition for Orlicz space to be fundamental.

Cohen elements in Banach algebras

SynopsisThe definition of Cohen elements in a commutative Banach algebra with a countable bounded approximate identity given by Esterle is modified slightly to be more analogous to the invertible elements in a unital Banach algebra. With the modified definition the n1-Cohen factorization results that were proved by Esterle are shown tohold in the semigroup of Cohen elements. If is the algebra of continuous complex valued functions vanishing at infinity on a σ-compact locally compact Hausdorff space X, then the Cohen elements in are identified and a natural quotient of a subsemigroup of Cohen elements is shown to be a group, isomorphic to the abstract index group of C(X∪{∞}).