The Tracial Topological Rank of C*-Algebras

Lin, Huaxin

doi:10.1112/plms/83.1.199

Cited by 147 publications

(191 citation statements)

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“…By Corollary 5.7 and Theorem 6.8 of [21], the order on projections over A is determined by traces, and by Theorem 3.4 of [20], the algebra A has real rank zero. So C * (Z, A, α) has real rank zero by Theorem 4.5.…”

Section: Real Rank Of Crossed Productsmentioning

confidence: 99%

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

Osaka¹,

Phillips²

2006

Ergod. Th. Dynam. Sys.

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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

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Section: Real Rank Of Crossed Productsmentioning

confidence: 99%

“…To prove that p 0 p 1 , we use the fact that, by Theorems 5.8 and 6.8 of [21], the order on projections over A is determined by traces. We certainly have…”

Section: Then Definementioning

confidence: 99%

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

Osaka¹,

Phillips²

2006

Ergod. Th. Dynam. Sys.

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“…Proof . It follows from Theorem 6.11 in [7] that K 0 (A) is a countable weakly unperforated simple (partial) ordered group with the Riesz interpolation property. Consequently, ρ A (K 0 (A)) is a countable unperforated simple ordered group with the Riesz interpolation property.…”

Section: Remark 35mentioning

confidence: 99%

“…Tracial topological rank for C * -algebras was introduced in [7] (see also [6]). It plays an important role in the study of classification of amenable C * -algebras.…”

Section: Introductionmentioning

confidence: 99%

Tracial Equivalence for $C^*$-Algebras and Orbit Equivalence for Minimal Dynamical Systems

Lin

2005

Proceedings of the Edinburgh Mathematical Society

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We introduce the notion of tracial equivalence for C * -algebras. Let A and B be two unital separable C * -algebras. If they are tracially equivalent, then there are two sequences of asymptotically multiplicative contractive completely positive linear maps φn : A → B and ψn : B → A with a tracial condition such that {φn • ψn} and {ψn • φn} are tracially approximately inner. Let A and B be two unital separable simple C * -algebras with tracial topological rank zero. It is proved that A and B are tracially equivalent if and only if A and B have order isomorphic ranges of tracial states. For the Cantor minimal systems (X 1 , σ 1 ) and (X 2 , σ 2 ), using a result of Giordano, Putnam and Skau, we show that two such dynamical systems are (topological) orbit equivalent if and only if the associated crossed products C(X 1 ) ×σ 1 Z and C(X 2 ) ×σ 2 Z are tracially equivalent.

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“…However the class of AFalgebras is much smaller than the class of exact C * -algebras. In [9], H. Lin defined a notion of topological rank for C * -algebras, which is called tracial rank. Roughly speaking, AF-algebras are C * -algebras that can be approximated in norm by finite dimensional C * -algebras.…”

Section: Introductionmentioning

confidence: 99%