Editing Von Neumann algebra (section)

==Amenable von Neumann algebras==
{{harvtxt|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 [[derivation (abstract algebra)|derivation]]s of ''M'' with values in a normal dual Banach bimodule are all inner.<ref>{{Cite journal|last=Connes|first=A|date=May 1978|title=On the cohomology of operator algebras |journal=Journal of Functional Analysis|language=en|volume=28|issue=2|pages=248–253|doi=10.1016/0022-1236(78)90088-5|doi-access=}}</ref>
*''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<sub>''n''</sub>, I<sub>∞</sub>, II<sub>1</sub>, II<sub>∞</sub>, III<sub>λ</sub>, for 0 < λ ≤ 1, and the ones of type III<sub>0</sub> correspond to certain ergodic flows. (For type III<sub>0</sub> 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 II<sub>1</sub> were classified by {{harvtxt|Murray|von Neumann|1943}}, and the remaining ones were classified by {{harvtxt|Connes|1976}}, except for the type III<sub>1</sub> case which was completed by Haagerup.

All amenable factors can be constructed using the '''[[crossed product|group-measure space construction]]''' of [[Francis Joseph Murray|Murray]] and [[John von Neumann|von Neumann]] for a single [[ergodic]] transformation. In fact they are precisely the factors arising as [[crossed product]]s by free ergodic actions of ''Z'' or ''Z/nZ'' on abelian von Neumann algebras ''L''<sup>∞</sup>(''X''). Type I factors occur when the [[measure space]] ''X'' is [[atom (measure theory)|atomic]] and the action transitive. When ''X'' is diffuse or [[atom (measure theory)|non-atomic]], it is [[equivalence (measure theory)|equivalent]] to [0,1] as a [[measure space]]. Type II factors occur when ''X'' admits an [[equivalence (measure theory)|equivalent]] finite (II<sub>1</sub>) or infinite (II<sub>∞</sub>) measure, invariant under an action of ''Z''. Type III factors occur in the remaining cases where there is no invariant measure, but only an [[quasi-invariant measure|invariant measure class]]: these factors are called '''Krieger factors'''.