Von Neumann Algebra - Amenable Von Neumann Algebras

Amenable Von Neumann Algebras

Connes (1976) and others proved that the following conditions on a von Neumann algebra M on a separable Hilbert space H are all equivalent:

• M is hyperfinite or AFD or approximately finite dimensional or approximately finite: this means the algebra contains an ascending sequence of finite dimensional subalgebras with dense union. (Warning: some authors use "hyperfinite" to mean "AFD and finite".)

• M is amenable: this means that the derivations of M with values in a normal dual Banach bimodule are all inner.

• M has Schwartz's property P: for any bounded operator T on H the weak operator closed convex hull of the elements uTu* contains an element commuting with M.

• M is semidiscrete: this means the identity map from M to M is a weak pointwise limit of completely positive maps of finite rank.

• M has property E or the Hakeda-Tomiyama extension property: this means that there is a projection of norm 1 from bounded operators on H to M '.

• M is injective: any completely positive linear map from any self adjoint closed subspace containing 1 of any unital C*-algebra A to M can be extended to a completely positive map from A to M.

There is no generally accepted term for the class of algebras above; Connes has suggested that amenable should be the standard term.

The amenable factors have been classified: there is a unique one of each of the types I_n, I_∞, II₁, II_∞, III_λ, for 0<λ≤ 1, and the ones of type III₀ correspond to certain ergodic flows. (For type III₀ calling this a classification is a little misleading, as it is known that there is no easy way to classify the corresponding ergodic flows.) The ones of type I and II₁ were classified by Murray & von Neumann (1943), and the remaining ones were classified by Connes (1976), except for the type III₁ case which was completed by Haagerup.

All amenable factors can be constructed using the group-measure space construction of Murray and von Neumann for a single ergodic transformation. In fact they are precisely the factors arising as crossed products by free ergodic actions of Z or Z_n on abelian von Neumann algebras L∞(X). Type I factors occur when the measure space X is atomic and the action transitive. When X is diffuse or non-atomic, it is equivalent to as a measure space. Type II factors occur when X admits an equivalent finite (II₁) or infinite (II_∞,) measure, invariant under an action of . Type III factors occur in the remaining cases where there is no invariant measure, but only an invariant measure class: these factors are called Krieger factors.