Measure continuous derivations on von Neumann algebras and applications to L^2-cohomology

Abstract

We prove that norm continuous derivations from a von Neumann algebra into the algebra of operators affiliated with its tensor square are automatically continuous for both the strong operator topology and the measure topology. Furthermore, we prove that the first continuous L^2-Betti number scales quadratically when passing to corner algebras and derive an upper bound given by Shen's generator invariant. This, in turn, yields vanishing of the first continuous L^2-Betti number for II_1 factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors with property Gamma.

17 pages

References in corpus (5)

• L^2-Homology for von Neumann Algebras
• L^2-Betti numbers of locally compact groups and their cross section equivalence relations
• L^2-Betti Numbers of Discrete Measured Groupoids
• Paths in quantum Cayley trees and L^2-cohomology
• Amenability and vanishing of L^2-Betti numbers: an operator algebraic approach

Cited by in corpus (3)

• On the first continuous $L^2$-cohomology of free group factors
• Vanishing of the first continuous L^2-cohomology for II_1 factors
• $L^2$-homology for inclusions of von Neumann algebras