Amenability and virtual diagonals for von Neumann algebras
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.
module actions are assumed \(\sigma\)-weak, weak* continuous
tensor product construction based on an assumed Grothendieck inequality
all nuclear \(C^*\)-algebras are amenable
scientific article; zbMATH DE number 4069599