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Representations of topological algebras by projective limits. (English) Zbl 1209.46023
Summary: 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 absorbent algebras (respectively, $A$-($k$-normed) algebras, where $k\in \left(0,1\right]$ 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 $k\in \left(0,1\right]$ varies), and every $m$-algebra is locally $m$-pseudoconvex. A condition for submultiplicativity of an $F$-seminorm is given.
##### MSC:
 46H05 General theory of topological algebras 46H20 Structure and classification of topological algebras