The Dixmier property and tracial states for C*-algebras

Robert Archbold; Leonel Robert; Aaron Tikuisis

doi:10.1016/j.jfa.2017.06.026

The Dixmier property and tracial states for C*-algebras

Robert Archbold, Leonel Robert, Aaron Tikuisis

Mathematical Science

University of Louisiana at Lafayette

Research output: Contribution to journal › Article › peer-review

13 Citations (Scopus)

12 Downloads (Pure)

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 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	https://doi.org/10.1016/j.jfa.2017.06.026
Publication status	Published - Oct 2017

Bibliographical note

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.

Keywords

C*-algebra
Dixmier property
tracial states
ultrapower

Access to Document

10.1016/j.jfa.2017.06.026Licence: CC BY

The Dixmier property and tracial states for C⁎-algebras
© 2017 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)
Final published version, 0.98 MBLicence: CC BY

Robert Archbold
- School of Natural & Computing Sciences, Mathematical Science - Emeritus Professor
Person: Honorary

Cite this

@article{26c65659008c41039f9d4efd53cb9c35,

title = "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 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.",

keywords = "C*-algebra, Dixmier property, tracial states, ultrapower",

author = "Robert Archbold and Leonel Robert and Aaron Tikuisis",

note = "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.",

year = "2017",

month = oct,

doi = "10.1016/j.jfa.2017.06.026",

language = "English",

volume = "273",

pages = "2655--2718",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Academic Press Inc.",

number = "8",

}

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

SN - 0022-1236

VL - 273

SP - 2655

EP - 2718

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 8

ER -

The Dixmier property and tracial states for C*-algebras

Abstract

Bibliographical note

Keywords

Access to Document

Fingerprint

Profiles

Robert Archbold

Cite this