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Journal of Operator Theory

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

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

Authors:  Vadim Alekseev (1) and David Kyed (2)
Author institution:(1) Mathematisches Institut, Georg-August-Universitaet Gottingen, Bunsenstrasse 3-5, D-37073 Goettingen, 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 $\twoone$ factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors with property Gamma.

Keywords: von Neumann algebras, $L^2$-Betti numbers, property $\T$

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