DIVISIBLE OPERATORS IN VON NEUMANN ALGEBRAS

Relativizing an idea from multiplicity theory, we say that an element of a von Neumann algebra M is n-divisible if it commutes with a type I-n subfactor. We decide the density of the n-divisible operators, for various n, M, and operator topologies. The most sensitive case is sigma-strong density in II1 factors, which is closely related to the McDuff property. We also make use of Voiculescu's noncommutative Weyl-von Neumann theorem to obtain several descriptions of the norm closure of the n-divisible operators in B(l(2)). Here are two consequences: (1) in contrast to the larger class of reducible operators, the divisible operators are nowhere dense; (2) if an operator is a norm limit of divisible operators, it is actually a norm limit of unitary conjugates of a single divisible operator. The following application is new even for B(l(2)): if an element of a von Neumann algebra belongs to the norm closure of the N-0-divisible operators, then the weak* closure of its unitary orbit is convex.