Divisible operators in von Neumann algebras

Abstract

Relativizing an idea from multiplicity theory, we say that an element of a von Neumann algebra $\mathcal{M}$ is $n$-divisible if it commutes with a type $\mathrm{I}_n$ subfactor. We decide the density of the $n$-divisible operators, for various $n$, $\mathcal{M}$, and operator topologies. The most sensitive case is $\sigma$-strong density in $\mathrm{II}_1$ 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 $\mathcal{B}(\ell^2)$. 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 $\mathcal{B}(\ell^2)$: if an element of a von Neumann algebra belongs to the norm closure of the $\aleph_0$-divisible operators, then the weak* closure of its unitary orbit is convex.