2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/162448Relativizing an idea from multiplicity theory, we say that an element x of a von Neumann algebra M is n-divisible if (W*(x)' cap M) unitally contains a factor of type I_n. 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 II_1 factors, which is closely related to the McDuff property. We 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(ell^2). Here are two consequences: (1) in contrast to the reducible operators, of which they form a subset, 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. This is related to our ongoing work on unitary orbits by the following theorem, which is new even for B(ell^2): if an element of a von Neumann algebra belongs to the norm closure of the aleph_0-divisible operators, then the sigma-weak closure of its unitary orbit is convex.37 pagesOperator AlgebrasFunctional Analysis47C15 (Primary); 47A65, 46L10 (Secondary)text