The Hochschild cohomology problem for von Neumann algebras

The hochschild cohomology problem for von neumann algebras.

In 1967, when Kadison and Ringrose began the development of continuous cohomology theory for operator algebras, they conjectured that the cohomology groups Hn(M, M), n >/= 1, for a von Neumann algebra M, should all be zero. This conjecture, which has important structural implications for von Neumann algebras, has been solved affirmatively in the type I, IIinfinity, and III cases, leaving open o...

A Survey of Hochschild Cohomology for von Neumann Algebras

In this paper we survey the field of Hochschild cohomology for von Neumann algebras. We describe the basic definitions and results without assuming background knowledge beyond some familiarity with von Neumann algebra theory. We offer no formal proofs of results, although we give detailed references to the literature. Instead, we concentrate on explaining the main techniques and how they are ap...

L-cohomology for Von Neumann Algebras

We study L-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [4]. We give a definition of Lcohomology and show how the study of the first L-Betti number can be related to the study of derivations with values in a bi-module of affiliated operators. We show several results about the possibility of extending derivations from sub-algebras and about uniqueness of...

Cohomology of Hopf C-algebras and Hopf von Neumann algebras

We will define two canonical cohomology theories for Hopf C∗-algebras and for Hopf von Neumann algebras (with coefficients in their comodules). We will then study the situations when these cohomologies vanish. The cases of locally compact groups and compact quantum groups will be considered in more details. 1991 AMS Mathematics Classification number: Primary: 46L55, 46L05; Secondary: 43A07, 22D25

Second Hochschild Cohomology of Convolution Algebras