L²-cohomology for von Neumann algebras

2006/01/18 by Thom, Andreas · 1 citation
#46L10 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.math/0601447

Abstract

We study L²-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes. We give a definition of L²-cohomology and show how the study of the first L²-Betti number can be related with 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 such extensions. Along the way, we prove some results about the dimension function of modules over rings of affiliated operators which are of independent interest.

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• Rank metric isometries and determinant-preserving mappings on II₁-factors

Related

• L²-Homology for von Neumann Algebras
• L²-homology for inclusions of von Neumann algebras
• Dimension theory of arbitrary modules over finite von Neumann algebras and applications to L²-Betti numbers
• L²-Betti numbers for subfactors
• Equivariant homologies for operator algebras