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Connes-amenability of $$WAP(\mathfrak B^*)^*$$. (English) Zbl 1419.46031
Summary: For a Banach algebra $$\mathfrak B$$, the set of weakly almost periodic functions on $$\mathfrak B$$ is denoted by $$WAP(\mathfrak B^*)$$. It is known that amenability of $$\mathfrak B$$ yields Connes-amenability of $$WAP(\mathfrak B^*)^*$$. The converse is not generally true though. We prove that under certain assumptions, $$\mathfrak B$$ is amenable if and only if $$WAP(\mathfrak B^*)^*$$ is Connes-amenable. As a result, we show that for a reflexive Banach space $$E$$ with the approximation property, $$K(E)$$ is amenable if and only if $$WAP(K(E)^*)^*$$ is Connes-amenable.
##### MSC:
 46H20 Structure, classification of topological algebras 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 47L10 Algebras of operators on Banach spaces and other topological linear spaces
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