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SMOOTH BIMODULES AND COHOMOLOGY OF II1 FACTORS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 16 / Issue 1 / February 2017
- Published online by Cambridge University Press:
- 07 May 2015, pp. 155-187
- Print publication:
- February 2017
-
- Article
- Export citation
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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.
