On the Dixmier property of simple C*-algebras

Abstract

A unital C*-algebra is said to satisfy the Dixmier property if for each element x in the closed convex hull of all elements of the form u*xu, u being a unitary in , intersects the centre of ((2), 2·7). The von Neumann algebras and also some other classes of C*-algebras are known to satisfy the Dixmier property (cf. (2), (3), (4), (6)). If is a simple C*-algebra which satisfies the Dixmier property then has at most one tracial state. In (3) Archbold raised the question whether there exists a unital simple C*-algebra which has at most one tracial state without satisfying the Dixmier property. In the present note we characterize the unital simple C*-algebras with at most one tracial state in terms of a condition which is similar to the Dixmier property, but is in fact formally weaker in the framework of simple C*-algebras. This characterization relies on the method used by Pedersen in (5) in order to show that for a unital simple C*-algebra which has at most one tracial state and at least one non-trivial projection the linear span of all projections in is dense in As an application we characterize those unital simple C*-algebras with a unique tracial state which satisfy the Dixmier property.

Closed convex hulls of unitary orbits in C⁎-algebras of real rank zero

“…Proof. The following argument can be found in [54] but is repeated for convenience of the reader. Suppose I is a non-zero ideal in A.…”

“…For example, the Dixmier property for a C * -algebra ( [18]) asks that the centre of the C * -algebra interests every such orbit. By [55], one need only consider self-adjoint operators to verify the Dixmier property and [26] (also see [54]) shows that a unital C * -algebra A has the Dixmier property if and only if A is simple and has at most one faithful tracial state.…”

Factoriality and Type Classification of k-Graph von Neumann Algebras

“…It follows from Proposition 4.6 that A has the weak Dixmier property. Therefore, A is simple and has the Dixmier property by [HZ84,Rie82] Proof. As in [Yan10, Proposition 5.4], it is easy to see that ω is a σ-KMS over O θ .…”

“…Clearly, the weak Dixmier property is weaker than the Dixmier property. It was proved in [29] that A satisfies the weak Dixmier property if and only if A is simple and has at most one tracial state. So it is now apparent that the Dixmier property and the weak Dixmier property are equivalent for simple C * -algebras.…”

“…Haagerup and Zsidó [HZ84] showed the converse for simple C*-algebras: every simple C*-algebra with at most one tracial state has the Dixmier property. The weak Dixmier property is introduced in [Rie82] in order to partially answer a question posed in [Arc79]. Clearly, the weak Dixmier property is weaker than the Dixmier property.…”