Smooth bimodules and cohomology of II$_1$ factors

TL;DR

This paper proves that under broad conditions, the first cohomology of a von Neumann algebra with coefficients in certain smooth bimodules vanishes, extending understanding of derivations and cohomology in operator algebras.

Contribution

It establishes the vanishing of 1-cohomology for von Neumann algebras with coefficients in smooth bimodules, generalizing previous results and including non-compact smooth elements.

Findings

Vanishing of 1-cohomology for von Neumann algebras with smooth bimodule coefficients

Inner derivations characterize all derivations into smooth bimodules

Existence of non-compact smooth elements in operator bimodules

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 $s^*$-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.