Published online by Cambridge University Press: 07 May 2015
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.
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