Finite cyclic group actions with the tracial Rokhlin property

Phillips, N.

doi:10.1090/tran/5566

Finite cyclic group actions with the tracial Rokhlin property
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Trans. Amer. Math. Soc. 367 (2015), 5271-5300 Request permission

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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