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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.

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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