  It is shown that a unital C*-algebra A has the Dixmier property if and only
if it is weakly central and satisfies certain tracial conditions. This
generalises the Haagerup-Zsido theorem for simple C*-algebras. We also study a
uniform version of the Dixmier property, as satisfied for example by von
Neumann algebras and the reduced C*-algebras of Powers groups, but not by all
C*-algebras with the Dixmier property, and we obtain necessary and sufficient
conditions for a simple unital C*-algebra with unique tracial state to have
this uniform property. We give further examples of C*-algebras with the uniform
Dixmier property, namely all C*-algebras with the Dixmier property and finite
radius of comparison-by-traces. Finally, we determine the distance between two
Dixmier sets, in an arbitrary unital C*-algebra, by a formula involving tracial
data and algebraic numerical ranges.
