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Amenability and weak amenability of second conjugate Banach algebras. (English) Zbl 0851.46035
Proc. Am. Math. Soc. 124, No. 5, 1489-1497 (1996); addendum ibid. 148, No. 10, 4573-4575 (2020).
Summary: For a Banach algebra \({\mathfrak A}\), amenability of \({\mathfrak A}^{**}\) necessitates amenability of \({\mathfrak A}\), and similarly for weak amenability provided \({\mathfrak A}\) is a left ideal in \({\mathfrak A}^{**}\). For \({\mathfrak G}\) a locally compact group, indeed more generally, \(L^1 ({\mathfrak G})^{**}\) is amenable if and only if \({\mathfrak G}\) is finite. If \(L^1 ({\mathfrak G})^{**}\) is weakly amenable, then \(M({\mathfrak G})\) is weakly amenable.

MSC:
46H20 Structure, classification of topological algebras
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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