The Dixmier property and tracial states for C*-algebras

Robert Archbold, Leonel Robert, Aaron Tikuisis

Research output: Contribution to journalArticle

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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–Zsid´o 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.
Original languageEnglish
Pages (from-to)2655-2718
Number of pages55
JournalJournal of Functional Analysis
Volume273
Issue number8
Early online date8 Jul 2017
DOIs
Publication statusPublished - Oct 2017

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C*-algebra
Unital
Simple C*-algebras
Numerical Range
Von Neumann Algebra
Radius
Trace
Necessary Conditions
Generalise
Sufficient Conditions
Arbitrary
Theorem

Keywords

  • C*-algebra
  • Dixmier property
  • tracial states
  • ultrapower

Cite this

The Dixmier property and tracial states for C*-algebras. / Archbold, Robert; Robert, Leonel; Tikuisis, Aaron.

In: Journal of Functional Analysis, Vol. 273, No. 8, 10.2017, p. 2655-2718.

Research output: Contribution to journalArticle

Archbold, Robert ; Robert, Leonel ; Tikuisis, Aaron. / The Dixmier property and tracial states for C*-algebras. In: Journal of Functional Analysis. 2017 ; Vol. 273, No. 8. pp. 2655-2718.
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