Abstract.
We study L 2-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [CoS]. We give a definition of L 2-cohomology and show how the study of the first L 2-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 such extensions. In particular, we show that the first L 2-Betti number of a tracial von Neumann algebra coincides with the corresponding number for an arbitrary weakly dense sub-C*-algebra.
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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Received: March 2006 Revision: October 2006 Accepted: October 2006
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Thom, A. L 2-Cohomology for Von Neumann Algebras. GAFA Geom. funct. anal. 18, 251–270 (2008). https://doi.org/10.1007/s00039-007-0634-7
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DOI: https://doi.org/10.1007/s00039-007-0634-7