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SMOOTH BIMODULES AND COHOMOLOGY OF II1 FACTORS

Published online by Cambridge University Press:  07 May 2015

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)
Corresponding

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.

Type
Research Article
Copyright
© Cambridge University Press 2015 

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