# Measure continuous derivations on von Neumann algebras and applications to L2-cohomology

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### Resumé

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 L2-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 L2-Betti number for \twoone factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors with property Gamma.
Originalsprog Engelsk Journal of Operator Theory 73 1 91-111 0379-4024 https://doi.org/10.7900/jot.2013sep23.2018 Udgivet - 1. jan. 2015

### Fingeraftryk

Von Neumann Algebra
Cohomology
Betti numbers
Strong Operator Topology
Algebra
Fundamental Group
Finitely Generated
Tensor
Generator
Upper bound
Topology
Norm
Invariant
Operator

### Citer dette

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title = "Measure continuous derivations on von Neumann algebras and applications to L2-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 L2-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 L2-Betti number for \twoone factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors with property Gamma.",
author = "David Kyed and Vadim Alekseev",
year = "2015",
month = "1",
day = "1",
doi = "10.7900/jot.2013sep23.2018",
language = "English",
volume = "73",
pages = "91--111",
journal = "Journal of Operator Theory",
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I: Journal of Operator Theory, Bind 73, Nr. 1, 01.01.2015, s. 91-111.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - Measure continuous derivations on von Neumann algebras and applications to L2-cohomology

AU - Kyed, David

PY - 2015/1/1

Y1 - 2015/1/1

N2 - 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 L2-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 L2-Betti number for \twoone factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors with property Gamma.

AB - 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 L2-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 L2-Betti number for \twoone factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors with property Gamma.

UR - http://www.mathjournals.org/jot/2015-073-001/

UR - https://arxiv.org/abs/1110.6155

U2 - 10.7900/jot.2013sep23.2018

DO - 10.7900/jot.2013sep23.2018

M3 - Journal article

VL - 73

SP - 91

EP - 111

JO - Journal of Operator Theory

JF - Journal of Operator Theory

SN - 0379-4024

IS - 1

ER -