  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.
