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