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Connes-amenability of multiplier Banach algebras. (English) Zbl 1201.46043
Let \(B\) be a Banach algebra with a bounded approximate identity, and let \(M(B)\) be its multiplier algebra under composition of maps with identity \(1_{M(B)}\). The authors prove that, if there exists a continuous linear injection \(B^* \rightarrow M(B)\) such that, for every \(u,v \in B^*\) and every \(b \in B,\) \(\langle u,vb \rangle _B = \langle v,bu \rangle _B\), then \(M(B)\) is a dual Banach algebra and the following are equivalent: 8mm
(i)
\(B\) is amenable;
(ii)
\(M(B)\) is Connes amenable;
(iii)
\(M(B)\) has a normal, virtual diagonal.
This main theorem improves Proposition 4.2 in [V. Runde, “Amenability for dual Banach algebras”, Stud. Math. 148, No. 1, 47–66 (2001; Zbl 1003.46028)].
MSC:
46H20 Structure, classification of topological algebras
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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