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–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.
Highlights
Let A be a unital C∗-algebra with unitary group U(A) and centre Z(A)
In [23], it was shown that every von Neumann algebra has the Dixmier property and an example was given of a unital C∗-algebra for which the Dixmier property does not hold
We show that any C∗-algebra with the Dixmier property and with finite radius of comparison-by-traces has the uniform Dixmier property (Corollary 3.22)
Summary
Let A be a unital C∗-algebra with unitary group U(A) and centre Z(A). For a ∈ A, the Dixmier set DA(a) is the norm-closed convex hull of the set {uau∗ : u ∈ U (A)}. The case of trivial centre in Theorem 1.1 is already an interesting generalisation of the Haagerup–Zsido theorem: a unital C∗-algebra A has the Dixmier property with centre Z(A) = C1 if and only if A has a unique maximal ideal J, A has at most one tracial state and J has no tracial states (Corollary 2.10) This result extends Theorem 1.2 in several ways: first by considering the Dixmier sets of a pair of elements a and b (rather than one of them being zero), second by providing a distance formula between these sets (rather than focusing on the case that this distance is zero), and third by allowing the elements a and b to be non-self-adjoint. We obtain elements in Z(A) by using Michael’s selection theorem, rather than the Katetov–Tong theorem (cf. [58], [90])
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