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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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