L^2-Homology for von Neumann Algebras

Alain Connes; Dimitri Shlyakhtenko

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arXiv · math/0309343

L^2-Homology for von Neumann Algebras

Alain Connes·Dimitri Shlyakhtenko

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Abstract

We define the notion of L^2 homology and L^2 Betti numbers for a tracial von Neumann algebra, or, more generally, for any involutive algebra with a trace. The definition of these invariants is obtained from the definition of L^2 homology for groups, using the ideas from the theory of correspondences. For the group algebra of a discrete group, our Betti numbers coincide with the L^2 Betti numbers of the group. We find a link between the first L^2 Betti number and free entropy dimension, which points to the non-vanishing of L^2 homology for the von Neumann algebra of a free group.

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BibTeX RIS

Alain Connes, Dimitri Shlyakhtenko. 2003-09-20. L^2-Homology for von Neumann Algebras. https://arxiv.org/abs/math/0309343

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