  The starting points for this paper are simple descriptions of the norm and
strong* closures of the unitary orbit of a normal operator in a von Neumann
algebra. The statements are in terms of spectral data and do not depend on the
type or cardinality of the algebra.
  We relate this to several known results and derive some consequences, of
which we list a few here. Exactly when the ambient von Neumann algebra is a
direct sum of sigma-finite algebras, any two normal operators have the same
norm-closed unitary orbit if and only if they have the same strong*-closed
unitary orbit if and only if they have the same strong-closed unitary orbit.
But these three closures generally differ from each other and from the unclosed
unitary orbit, and we characterize when equality holds between any two of these
four sets. We also show that in a properly infinite von Neumann algebra, the
strong-closed unitary orbit of any operator, not necessarily normal, meets the
center in the (non-void) left essential central spectrum of Halpern. One
corollary is a "type III Weyl-von Neumann-Berg theorem" involving containment
of essential central spectra.
