Finite cyclic group actions with the tracial Rokhlin property

Abstract

We give examples of actions of Z/2Z 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 Z/2Z 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 Z/2Z 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 Z/2Z 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 Z/2Z with the tracial Rokhlin property need not be AF. We give examples of a tracially approximately representable action of Z/2Z on a simple unital AF algebra which is nontrivial on K_0, and of a tracially approximately representable action of Z/2Z on a simple unital AT algebra with real rank zero which is nontrivial on K_1.