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Amenability for dual Banach algebras. (English) Zbl 1003.46028
A Banach algebra $${\mathfrak A}$$ is said to be dual if $${\mathfrak A}=({\mathfrak A}_*)^*$$ for a closed submodul $${\mathfrak A}_*$$ of $${\mathfrak A}^*$$. Conditions for amenable dual Banach algebras to imply that they are finite direct sums of full matrix algebras are presented. As the revealing property of amenability is rarely shared by dual algebras and in order to get a more extensive area for investigation the conception of amenability is modified in the following way: $${\mathfrak A}$$ is Connes amenable if $${\mathcal H}^1_{w^*}({\mathfrak A}, E^*)= \{0\}$$ for every $$w^*$$-Banach $${\mathfrak A}$$-bimodule $$E^*$$ (i.e. the module multiplication is $$w^*$$-continuous with respect to $${\mathfrak A}_*$$) where the first cohomology group is formed in terms of the space of all $$w^*$$-continuous derivatives from $${\mathfrak A}$$ into $$E^*$$. This additionally is refined by the notion of strong Connes amenability. It is shown that an Arens regular Banach algebra $${\mathfrak A}$$ which is an ideal in $${\mathfrak A}^{**}$$ is amenable if and only if $${\mathfrak A}^{**}$$ is Connes amenable. The relations of strong Connes amenability to amenability of special classes of Banach algebras are established. The Connes amenability of some dual Banach algebras occurring in abstract harmonic analysis is treated, and a new proof for a characterization of amenable $$W^*$$-algebras is accomplished.

##### MSC:
 46H05 General theory of topological algebras 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 43A10 Measure algebras on groups, semigroups, etc. 46B28 Spaces of operators; tensor products; approximation properties 46L10 General theory of von Neumann algebras 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
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