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 In, I∞, II1, II∞, IIIλ, 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 Zn 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.
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