Triple derivations on von Neumann algebras
It is well known that every derivation of a von Neumann
algebra into itself is an inner derivation
and that every derivation of a von Neumann algebra into its predual is
inner. It is less well known that every triple derivation of a von
Neumann algebra into itself is an inner triple derivation. We examine to
what extent all triple derivations of a von Neumann algebra into its
predual are inner. This rarely happens but it comes close.
We prove a (triple) cohomological characterization of finite factors
and a zero-one law for factors.
Namely, we show that for any factor, the linear space of triple
derivations into the predual, modulo the norm closure of the inner
triple derivations, has dimension 0 or 1: It is zero if and only if the
factor is finite.
Robert Pluta < plutar@tcd.ie >
Bernard Russo < brusso@uci.edu >