L^2-cohomology for von Neumann algebras

Details L^2-cohomology for von Neumann algebras L^2-cohomology for von Neumann algebras

2006-01-18

arxiv.org/abs/math/0601447v2

Mathematics

Operator Algebras

Scientific paper

We study L^2-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes. We give a definition of L^2-cohomology and show how the study of the first L^2-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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