Journal of Operator Theory

Volume 80, Issue 1, Summer 2018 pp. 187-211.

Maximally unitarily mixed states on a $C^*$-algebra

Authors: Robert Archbold (1), Leonel Robert (2), and Aaron Tikuisis (3)
Author institution: (1) Institute of Mathematics, University of Aberdeen, King's College, Aberdeen AB24 3UE, Scotland, U.K.
(2) Department of Mathematics, University of Louisiana at Lafayette, Lafayette, 70504-3568, U.S.A.
(3) Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON K1N 6N5, Canada

Summary: We investigate the set of maximally mixed states of a $C^*$-algebra, extending previous work by Alberti on von Neumann algebras. We show that, unlike for von Neumann algebras, the set of maximally mixed states of a $C^*$-algebra may fail to be weak* closed. We obtain, however, a concrete description of the weak* closure of this set, in terms of tracial states and states which factor through simple traceless quotients. For $C^*$-algebras with the Dixmier property or with Hausdorff primitive spectrum we are able to advance our investigations further. We pose several questions.

DOI: http://dx.doi.org/10.7900/jot.2017sep24.2168
Keywords: states of $C^*$-algebras, unitary mixings, Dixmier property

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