The Dixmier property and tracial states for C⁎-algebras

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-Zsidó 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.