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A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal. (English) Zbl 1098.46037

Trans. Am. Math. Soc. 358, No. 1, 391-402 (2006); erratum ibid. 367, No. 1, 751-754 (2015).
Let \(G\) be a locally compact group and let \({\mathcal WAP}(G)\) be the space of weakly almost periodic functions on \(G\). The author shows that if \(G\) is a non-compact [SIN]-group, then its dual Banach algebra \({\mathcal WAP}(G)^*\) does not have a normal, virtual diagonal. As a corollary, for a [SIN]-group \(G\), the property that \({\mathcal WAP}(G)^*\) has a normal, virtual diagonal is equivalent to the compactness of \(G\) and also \(G\) is amenable if and only if \({\mathcal WAP}(G)^*\) is Connes-amenable. If however \(G\) is locally compact minimally weakly almost periodic, then the author shows that \(G\) is amenable if and only if \({\mathcal WAP}(G)^*\) has a normal virtual diagonal. As an example the group \(G:=\mathbb R^N\rtimes SO(N)\) is minimally weakly almost periodic, so there exist non compact groups for which the algebra \({\mathcal WAP}(G)^*\) has a normal, virtual diagonal.
Reviewer: Jean Ludwig (Metz)

MSC:

46H20 Structure, classification of topological algebras
43A10 Measure algebras on groups, semigroups, etc.
22A15 Structure of topological semigroups
22A20 Analysis on topological semigroups
43A07 Means on groups, semigroups, etc.; amenable groups
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
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