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 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.

55 pages. Added Section 3.3 on explicit constants for the uniform Dixmier property, with some knock-on effects elsewhere (addition of Lemma 1.5, Theorem 1.6, Example 3.8, Lemma 3.9). Small corrections and clarifications made in a number of places. To appear in J. Funct. Anal

References in corpus (16)

• Quasitraces on exact C*-algebras are traces
• Boundaries of reduced C*-algebras of discrete groups
• The nuclear dimension of C*-algebras
• Nuclear dimension and n-comparison
• An intrinsic characterization of C*-simplicity
• Decomposition rank and Z-stability
• Model theory of ${\mathrm{C}}^{\ast }$-algebras
• A new look at C*-simplicity and the unique trace property of a group
• The kernel of the determinant map on pure C*-algebras
• Closed Convex Hulls of Unitary Orbits in C${}^{\ast }$-Algebras of Real Rank Zero
• C*-simplicity and the amenable radical
• Sums of commutators in pure C*-algebras
• Supramenable groups and partial actions
• Norms of inner derivations for multiplier algebras of C*-algebras and group C*-algebras, II
• Closed convex hulls of unitary orbits in certain simple real rank zero C${}^{\ast }$-algebras
• Unitary orbits of self-adjoint operators in simple Z-stable C*-algebras

Cited by in corpus (4)

• Centrally Stable Algebras
• On closed Lie ideals of certain tensor products of C*-algebras II
• The centre-quotient property and weak centrality for $C^{\ast }$-algebras
• Closed ideals and Lie ideals of minimal tensor product of certain C*-algebras