Smooth bimodules and cohomology of II${}_1$ factors

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 $\Cal H$ and $\Cal B_0\subset \Cal B(\Cal H)$ is a norm closed $M$-bimodule such that any $T\in \Cal B_0$ is {\it smooth} (i.e. the left and right multiplication of $T$ by $x\in M$ are continuous from the unit ball of $M$ with the $s^{\ast }$-topology to $\Cal B_0$ with its norm), then any derivation of $M$ into $\Cal B_0$ is inner. The compact operators are smooth over any $M\subset \Cal B(\Cal H)$, but there is a large variety of non-compact smooth elements as well.

36 pages

Cited by in corpus (2)

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