Abstract. We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Let A be a stably finite simple unital C*-algebra, and let α be an automorphism of A which has the tracial Rokhlin property. Suppose A has real rank zero and stable rank one, and suppose that the order on projections over A is determined by traces. Then the crossed product algebra C * (Z, A, α) also has these three properties.We also present examples of C*-algebras A with automorphisms α which satisfy the above assumptions, but such that C * (Z, A, α) does not have tracial rank zero.
IntroductionWe introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties which have appeared in the literature, such as in [12], [18], and [13], at least for C*-algebras which are tracially AF in the sense of [20], in roughly the same way that being tracially AF is weaker than the local characterization of AF algebras (Theorem 2.2 of [4]).Our main results are as follows. Let A be a stably finite simple unital C*-algebra, and let α be an automorphism of A which has the tracial Rokhlin property. Suppose A has real rank zero and stable rank one, and suppose that the order on projections over A is determined by traces (Blackadar's Second Fundamental Comparability Question, 1.3.1 of [2], for M ∞ (A)). Then C * (Z, A, α) also has these three properties. In fact, we will see that not all the hypotheses on A are needed for all the conclusions.The proofs are adapted from [27]. The arguments here are more difficult for several reasons. First, in [27] there is a single "large" AF subalgebra, and the properties of the reduced groupoid C*-algebra are obtained by comparison with this subalgebra. In this paper, we are not able to choose nested approximating subalgebras; moreover, even if we were, the direct limit would not be AF. Second, we assume that the order on projections over A is determined by all traces, but only the invariant traces extend to the crossed product. Third, in [27] we relied