Journal of Operator Theory

Volume 73, Issue 1, Summer 2015 pp. 91-111.

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

Authors: (1) Vadim Alekseev, (2) David Kyed
Author institution: (1) Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstraße 3-5, D-37073 Göttingen, Germany
(2) Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark

Summary: 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 $\rm II_1$ factors with property $(\rm T)$, for finitely generated factors with non-trivial fundamental group and for factors with property Gamma.

DOI: http://dx.doi.org/10.7900/jot.2013sep23.2018
Keywords: Von Neumann algebras, $L^2$-Betti numbers, property $(\rm T)$.

Contents Full-Text PDF