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SMOOTH BIMODULES AND COHOMOLOGY OF II1 FACTORS

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  07 May 2015

Alin Galatan  and
Sorin Popa
Alin Galatan
Affiliation:
Department of Mathematics, University of California, UCLA, Los Angeles, CA 90095-1555, USA (agalatan@math.ucla.edu; popa@math.ucla.edu)
Sorin Popa
Affiliation:
Department of Mathematics, University of California, UCLA, Los Angeles, CA 90095-1555, USA (agalatan@math.ucla.edu; popa@math.ucla.edu)
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Abstract

We prove that, under rather general conditions, the 1-cohomology of a von Neumann algebra $M$ with values in a Banach $M$-bimodule satisfying a combination of smoothness and operatorial conditions vanishes. For instance, we show that, if $M$ acts normally on a Hilbert space ${\mathcal{H}}$ and ${\mathcal{B}}_{0}\subset {\mathcal{B}}({\mathcal{H}})$ is a norm closed $M$-bimodule such that any $T\in {\mathcal{B}}_{0}$ is smooth (i.e., the left and right multiplications of $T$ by $x\in M$ are continuous from the unit ball of $M$ with the $s^{\ast }$-topology to ${\mathcal{B}}_{0}$ with its norm), then any derivation of $M$ into ${\mathcal{B}}_{0}$ is inner. The compact operators are smooth over any $M\subset {\mathcal{B}}({\mathcal{H}})$, but there is a large variety of non-compact smooth elements as well.

Keywords

von Neumann II1 factorscohomologyderivationssmooth bimodules

MSC classification

Primary: 46L10: General theory of von Neumann algebras 46L57: Derivations, dissipations and positive semigroups in $C^*$-algebras
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 1 , February 2017 , pp. 155 - 187
Copyright
© Cambridge University Press 2015 

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