Some results and criteria of existence concerning bounded approximate identities in Banach algebras are extended to the topological algebras setting. We thereby prove that the bidual of a commutative locally C*-algebra with either of the two Arens products is a unital commutative algebra, and that a quasinormable Fréchet m-convex algebra has a left (resp. right) bounded approximate identity if and only if it can be represented as an inverse limit of Banach algebras each of which has a left (resp. right) bounded approximate identity.
"Criteria of existence of bounded approximate identities in topological algebras." Bull. Belg. Math. Soc. Simon Stevin 17 (5) 949 - 960, december 2010. https://doi.org/10.36045/bbms/1292334069