The Dixmier property and tracial states for C⁎-algebras

Abstract

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