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Finite cyclic group actions with the tracial Rokhlin property


Author: N. Christopher Phillips
Journal: Trans. Amer. Math. Soc. 367 (2015), 5271-5300
MSC (2010): Primary 46L55; Secondary 46L40
Published electronically: April 7, 2015
MathSciNet review: 3347172
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Abstract: We give examples of actions of $ \mathbb{Z} / 2 \mathbb{Z}$ on AF algebras and AT algebras which demonstrate the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability and tracial approximate representability. Specific results include the following. We determine exactly when a product type action of $ \mathbb{Z} / 2 \mathbb{Z}$ on a UHF algebra has the tracial Rokhlin property; in particular, unlike for the strict Rokhlin property, every UHF algebra admits such an action. We prove that Blackadar's action of $ \mathbb{Z} / 2 \mathbb{Z}$ on the $ 2^{\infty }$ UHF algebra, whose crossed product is not AF because it has nontrivial $ K_1$-group, has the tracial Rokhlin property, and we give an example of an action of $ \mathbb{Z} / 2 \mathbb{Z}$ on a simple unital AF algebra which has the tracial Rokhlin property and such that the $ K_0$-group of the crossed product has torsion. In particular, the crossed product of a simple unital AF algebra by an action of $ \mathbb{Z} / 2 \mathbb{Z}$ with the tracial Rokhlin property need not be AF. We give an example of a strongly selfabsorbing C*-algebra $ D$ (the $ 3^{\infty }$ UHF algebra), a $ D$-stable simple separable unital C*-algebra $ B,$ and an action of $ \mathbb{Z} / 2 \mathbb{Z}$ on $ B$ with the tracial Rokhlin property such that the crossed product is not $ D$-stable. We also give examples of a tracially approximately representable action of $ \mathbb{Z} / 2 \mathbb{Z}$ on a simple unital AF algebra which is nontrivial on $ K_0,$ and of a tracially approximately representable action of $ \mathbb{Z} / 2 \mathbb{Z}$ on a simple unital AT algebra with real rank zero which is nontrivial on $ K_1.$


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Additional Information

N. Christopher Phillips
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

DOI: https://doi.org/10.1090/tran/5566
Received by editor(s): December 30, 2007
Received by editor(s) in revised form: January 29, 2010
Published electronically: April 7, 2015
Additional Notes: This research was partially supported by NSF grants DMS 0070776 and DMS 0302401.
Article copyright: © Copyright 2015 American Mathematical Society