Journal of Operator Theory

Volume 94, Issue 2, Autumn 2025 pp. 293-337.

Compact group actions with the tracial Rokhlin property. I: Permanence properties

Authors: Javad Mohammadkarimi (1), N. Christopher Phillips (2)
Author institution: (1) School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education) and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China
(2) Department of Mathematics, University of Oregon, Eugene OR 97403-1222, U.S.A.

Summary: We define a ``tracial'' analog of the Rokhlin property for actions of second countable compact groups on infinite dimensional simple separable unital $C^*$-algebras. We prove that fixed point algebras under such actions (and, in appropriate cases, crossed products by such actions) preserve simplicity, Property~(SP), tracial rank zero, tracial rank at most one, the Popa property, tracial ${\mathcal{Z}}$-stability, ${\mathcal{Z}}$-stability when the algebra is nuclear, infiniteness, and pure infiniteness. We also show that the radius of comparison of the fixed point algebra is at most that of the original algebra. Examples, nonexistence theorems, and relations with other properties will appear elsewhere.

DOI: http://dx.doi.org/10.7900/jot.2023apr20.2489
Keywords: compact group action, tracial Rokhlin property, crossed product, fixed point algebra, permanence properties

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