### Abstract

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 language | English |
---|---|

Pages (from-to) | 2655-2718 |

Number of pages | 55 |

Journal | Journal of Functional Analysis |

Volume | 273 |

Issue number | 8 |

Early online date | 8 Jul 2017 |

DOIs | |

Publication status | Published - Oct 2017 |

### Fingerprint

### Keywords

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

### Cite this

*Journal of Functional Analysis*,

*273*(8), 2655-2718. https://doi.org/10.1016/j.jfa.2017.06.026

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

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 273, no. 8, pp. 2655-2718. https://doi.org/10.1016/j.jfa.2017.06.026

}

TY - JOUR

T1 - The Dixmier property and tracial states for C*-algebras

AU - Archbold, Robert

AU - Robert, Leonel

AU - Tikuisis, Aaron

N1 - A.T. was partially supported by an NSERC Postdoctoral Fellowship and through the EPSRC grant EP/N00874X/1. Acknowledgements We are grateful to Luis Santiago for helpful discussions at an early stage of this investigation. We would also like to thank the referee for providing helpful comments, which have led to a number of improvements.

PY - 2017/10

Y1 - 2017/10

N2 - It is shown that a unital C∗-algebra A has the Dixmier property if and only ifit 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 uniformDixmier 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.

AB - It is shown that a unital C∗-algebra A has the Dixmier property if and only ifit 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 uniformDixmier 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.

KW - C-algebra

KW - Dixmier property

KW - tracial states

KW - ultrapower

U2 - 10.1016/j.jfa.2017.06.026

DO - 10.1016/j.jfa.2017.06.026

M3 - Article

VL - 273

SP - 2655

EP - 2718

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 8

ER -