  We study variants of the Dixmier property that apply to elements of a unital
C*-algebra, rather than to the C*-algebra itself. By a Dixmier element in a
C*-algebra we understand one that can be averaged into a central element by
means of a sequence of unitary mixing operators. Examples include all
self-commutators and all quasinilpotent elements. We do a parallel study of an
element-wise version of weak centrality, where the averaging to the centre is
done using unital completely positive elementary operators (as in Magajna's
characterization of weak centrality). We also obtain complete descriptions of
more tractable sets of elements, where the corresponding averaging can be done
arbitrarily close to the centre. This is achieved through several "spectral
conditions", involving numerical ranges and tracial states.
