Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property

Abstract

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

Crossed products by finite group actions with the Rokhlin property

“…Compare with the results of [24] for crossed products by Z, and analogous results of [1] for the case of finite groups acting on simple C*-algebras. However, the proposition fails if no condition at all is put on the action.…”

The tracial Rokhlin property for automorphisms on non-simple C*-algebras

“…When A is a unital simple C*-algebra, the tracial Rokhlin property of the above definition is weaker than the Rokhlin property as in [10,12], we weak the condition (4) to only require that the positive element 1 − e can be compared with the given positive element a by the action of α.…”

“…It was adopted by Herman and Oeneanu [2] for UHF-algebras. Rørdam [13] and Kishimoto [6] introduced the Rokhlin property to a much more general context of C*-algebras, then Osaka and Phillips studied integer group actions which satisfy certain type of Rokhlin property on some simple C*-algebras (see [12]). More recently, Lin studied the Rokhlin property for automorphisms on simple C*-algebras (see [10]).…”

Certain Aperiodic Automorphisms of Unital Simple Projectionless C*-Algebras

“…Let A be a separable unital C * -algebra. We say that the order on projections over A is determined by traces if p, q ∈ P (M ∞ (A)) with τ (p) < τ (q) for any τ ∈ T (A) satisfy p q (i.e., there exists w ∈ A such that w * w = p and ww * ≤ q), see [1] and [16]. We say that A has the property (SI), if A satisfies that: if a unital C * -algebra B of real rank zero is given such that there exists a unital embedding from A into B and the order on projections over B is determined by traces, and any sequences (e n ) n , (f n ) n ∈ P (B ∞ ∩ A ′ ) with e n , f n ∈ B sa satisfy the following conditions:…”

“…On the assumption for the crossed product in the above proposition, Osaka and Phillips showed in [16] that: if A is an infinite dimensional stably finite simple unital C * -algebra of real rank zero, the order on projections over A is determined by traces, and α ∈ Aut(A) has the tracial Rohlin property, then the order on projections over A × α Z is determined by traces.…”