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Criteria of existence of bounded approximate identities in topological algebras. (English) Zbl 1214.46029

Some results known for Banach algebras are generalized to the general topological algebra setting or to the hypotopological algebra setting. Criteria for a topological algebra to have a left approximate identity are proven. Criteria for a right hypotopological algebra to possess left bounded approximate units are given. Some properties concerning ideals of a topological algebra, having bounded approximate identities, are considered. It is proved that a quasinormable Fréchet \(m\)-convex algebra \(A\) has a left bounded approximate identity if and only if it has an Arens-Michael reperesentation \(A=\varprojlim A_n\), where each \(A_n\) has a left bounded approximate identity.
Reviewer: Mart Abel (Tartu)

MSC:

46H20 Structure, classification of topological algebras
46A20 Duality theory for topological vector spaces
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46K05 General theory of topological algebras with involution
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
46M40 Inductive and projective limits in functional analysis
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Full Text: Euclid