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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 $$G$$.

##### MSC:
 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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