  Relativizing 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.
