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Connes-amenability, super-amenability and normal, virtual diagonal for some Banach algebras. (English) Zbl 1128.43003

A Banach algebra \(A\) is called super-amenable (or contractible) if every bounded derivation from \(A\) into a Banach \(A\)-bimodule is inner. Examples of super-amenable Banach algebras are rare: the only known ones are the classically semisimple ones, and it has been known for a long time that any super-amenable Banach algebra with the approximation property has to be classically semisimple [see, e.g., V. Runde, Lectures on amenability. Lecture Notes in Mathematics. 1774. Berlin: Springer. (2002; Zbl 0999.46022)]. Consequently, whenever \(G\) is a compact group such that, for \(p\in(1,\infty)\), \(L^p(G)\) equipped with the convolution product is super-amenable, the group \(G\) has to be finite: this is the first of the main results of the paper under review. The author then proceeds to give a slightly different proof of a result by the reviewer [Math. Scand. 95, No. 1, 124–144 (2004; Zbl 1087.46035)], namely that a locally compact \(G\) is amenable if and only if the dual Banach algebra \(\text{ WAP}(G)^*\) is Connes-amenable. (Like the reviewer’s proof, the given one still relies crucially on the main result of [V. Runde, J. Lond. Math. Soc. (2) 67, No. 3, 643–656 (2003; Zbl 1040.22002)].) Finally, the author extends another result by the reviewer [Bull. Aust. Math. Soc. 68, No. 2, 325–328 (2003; Zbl 1042.22001)] from measure algebras to weighted measure algebras: Let \(G\) be an amenable, locally compact group, and let \(\omega\) be a diagonally bounded weight on \(G\), i.e., the weight \(\Omega(x) := \omega(x)\omega(x^{-1})\) is bounded, then the dual Banach algebra \(M(G,\Omega)\) has a normal virtual diagonal.

MSC:

43A05 Measures on groups and semigroups, etc.
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46H20 Structure, classification of topological algebras
43A10 Measure algebras on groups, semigroups, etc.
22D15 Group algebras of locally compact groups
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