L^2-Homology for von Neumann Algebras

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.

References in corpus (2)

• On a Class of Type II${}_1$ Factors with Betti Numbers Invariants
• Some estimates for non-microstates free entropy dimension, with applications to $q$-semicircular families

Cited by in corpus (10)

• Free Entropy Dimension in Amalgamated Free Products
• Homology of free quantum groups
• Orthogonal free quantum group factors are strongly 1-bounded
• The free field: realization via unbounded operators and Atiyah property
• Measure continuous derivations on von Neumann algebras and applications to L^2-cohomology
• Finite Free Entropy and Free Group Factors
• On the first continuous $L^2$-cohomology of free group factors
• Amenability and vanishing of L^2-Betti numbers: an operator algebraic approach
• A characterization of rationality for operators in free semicircular elements
• L^2-cohomology for von Neumann algebras