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Connes-amenability and normal, virtual diagonals for measure algebras. I. (English) Zbl 1040.22002
It turns out that Connes-amenability is the “right” version of amenability for von Neumann algebras and is equivalent to several other important properties. But the definition of Connes amenability makes sense for a larger class of Banach algebras, which sometimes are called dual Banach algebras. The dual Banach algebra to be concerned with in the paper under review is the measure algebra $$M(G)$$ of a locally compact group $$G.$$ The main result of the paper is that if $$G$$ is amenable, $$M(G)$$ is Connes-amenable. The converse is also true and is an easy result.

##### MSC:
 22D15 Group algebras of locally compact groups 43A20 $$L^1$$-algebras on groups, semigroups, etc. 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 46E15 Banach spaces of continuous, differentiable or analytic functions 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 47B47 Commutators, derivations, elementary operators, etc. 43A10 Measure algebras on groups, semigroups, etc.
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