Property Gamma, Cohomology, Complemented Subspaces, Similarities and Length.
by
Erik Christensen
University of Copenhagen
Coauthors: Florin Pop (Wagner College), Alaan M. Sinclair (University of Edinburgh), Roger R. Smith (Texas A&M University)

Property Gamma, Cohomology, Complemented Subspaces, Similarities and Length.

F. J. Murray and J. von Neumann introduced a concept called property Gamma for von Neumann algebras in order to show that there are at least 2 non isomorphic von Neumann algebra factors which are generated by left regular representations of discrete groups.

The typical example of a von Neumann algebra which does not have the property Gamma is the von Neumann algebra generated by the free group on 2 generators. The von Neumann algebras with property Gamma have a certain commutation property which can be used to show that these algebras have a lot of properties, which we do not know if all von Neumann algebras have.

Based on some recent results of Pisier concerning the length of a von Neumann algebra with property Gamma, we can push the validity of several results exactly to the Gamma case and not further.

The results are all related to the concept of completely bounded mappings and in general one can say that the property Gamma helps to establish the necessary complete boundedness.

The results obtained deals with the following questions:

The vanishing of Hochshild cohomology groups for von Neumann algebras.

If a von Neumann algebra is a complemented subspace of B(H), is it then injective ?

What are the possible lengths of a C*-algebra ?

Date received: June 13, 2002