Representations of topological algebras by projective limits

Abstract

It is shown that a) it is possible to define the topology of any topological algebra by a collection of $F$-seminorms, b) every complete locally uniformly absorbent (complete locally $A$-pseudoconvex) Hausdorff algebra is topologically isomorphic to a projective limit of metrizable locally uniformly\break absorbent algebras (respectively, $A$-($k$-normed) algebras, where $k\in(0,1]$ varies, c) every complete locally idempotent (complete locally $m$-pseudoconvex) Hausdorff algebra is topologically isomorphic to a projective limit of locally idempotent Fréchet algebras (respectively, $k$-Banach algebras, where $k\in(0,1]$ varies) and every $m$-algebra is locally $m$-pseudoconvex. Condition for submultiplicativity of $F$-seminorm is given.