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Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule. (English) Zbl 1087.46035
Let \(A\) be a Banach algebra. \(A\) is called a dual Banach algebra if it is a dual Banach \(A\)-module. Dual Banach algebras include every von Neumann algebra and the measure algebra \(M(G)\) of a locally compact group \(G\). A dual Banach algebra \(A\) is called Connes-amenable if every weak\(^*\)-continuous derivation from \(A\) into a normal, dual Banach \(A\)-bimodule is inner. Connes-amenability was introduced by B. E. Johnson, R. V. Kadison and J. Ringrose for von Neumann algebras [Bull. Soc. Math. Fr. 100, 73–96 (1972; Zbl 0234.46066)]. Connes-amenability is equivalent to injectivity and semidiscreteness for von Neumann algebras.
For the measure algebra \(M(G)\), it was shown by the author that the measure algebra \(M(G)\) is Connes-amenable if and only if \(G\) is compact [J. Lond. Math. Soc., II. Ser. 67, No. 3, 643–656 (2003; Zbl 1040.22002)]. In the present paper, the author considers the following three properties of a dual Banach algebra: (i) \(A\) is Connes-amenable; (ii) \(A\) has a normal, virtual diagonal; (iii) \(A_*\) is an injective \(A\)-bimodule. He shows that (iii) implies (ii) and thus (i). However, the converse need not hold in general. This is done by considering the measure algebra \(M(G)\) of an infinite amenable locally compact group \(G\). It answers a question of A. Ya. Helemskii. These conditions are also studied for the Fourier Stieltjes algebra \(B(G)\) for certain \(G\).

46H05 General theory of topological algebras
43A10 Measure algebras on groups, semigroups, etc.
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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