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

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 $\text{Cal}H$ and $\Cal B_0\subset \Cal B(\Cal H)$isanormclosed$M$-bimodulesuchthatany$T\in \Cal B_0$issmooth(i\mathrm{.}e\mathrm{.}theleftandrightmultiplicationof$T$by$x\in M$arecontinuousfromtheunitballof$M$withthe$s^*$-topologyto$\Cal B_0$withitsnorm),thenanyderivationof$M$into$\Cal B_0$ is inner. The compact operators are smooth over any $M\subset \text{Cal}B(\text{Cal}H)$, but there is a large variety of non-compact smooth elements as well.