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Amenability and virtual diagonals for von Neumann algebras. (English) Zbl 0655.46053
Bimodules are defined for pairs of unital Banach algebras, and the module elements form a Banach space. If this space is a dual Banach space $$V^*$$ then the module is called a normal module when the algebras are von Neumann algebras, and the module actions are assumed $$\sigma$$-weak, weak* continuous.
A unital Banach algebra A is said to be amenable if, for any dual A bimodule $$V^*$$, any derivation $$\delta:A\to V^*$$ is inner. A von Neumann algebra R is amenable if this condition is satisfied for all normal R-bimodules and derivations $$R\to V^*$$. A normal virtual diagonal for R is an element $$M\in R{\hat \otimes}^{\sigma}R$$ such that $$rM=Mr$$, $$\forall r\in R$$, and $$\pi (M)=1$$ where $$\pi$$ is the restriction to $$R\otimes R$$ of a certain mapping $$\pi^*_ 0:R{\hat \otimes}_ 0^{\sigma}R\to R$$ associated to a certain tensor product construction based on an assumed Grothendieck inequality.
It is proved that a von Neumann algebra is amenable iff it has a normal virtual diagonal, and, as a corollary, that all nuclear $$C^*$$-algebras are amenable. The corollary was proved earlier by Haagerup. But the present paper yields a direct proof of Haagerup’s result, the basic idea being that a certain universal bimodule over a given von Neumann algebra is automatically normal.
Reviewer: P.E.T.Jørgensen

##### MSC:
 46L55 Noncommutative dynamical systems 46L10 General theory of von Neumann algebras 46L40 Automorphisms of selfadjoint operator algebras
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