Approximately central matrix units and the structure of noncommutative tori

Blackadar, Bruce; Kumjian, Alex; Rørdam, Mikael

doi:10.1007/bf00961466

Cited by 145 publications

(221 citation statements)

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“…If A has tracial rank zero, then RR(A) = 0 by Theorem 3.4 of [20], and the order on projections over A is determined by traces by Theorems 5.8 and 6.8 of [21]. Under the other hypotheses, these conclusions follow from Corollary 3.9(b) and Theorem 1.4(e) of [3]. Accordingly, we verify the conditions of Lemma 1.4.…”

Section: Then Definesupporting

confidence: 67%

“…• A is approximately divisible in the sense of [3], all quasitraces on A are traces, and projections in A distinguish the traces on A.…”

Section: The Tracial Rokhlin Propertymentioning

confidence: 99%

“…It follows from Corollary 2.10 of [3] that there is a finite dimensional unital subalgebra E ⊂ A such that [a, x] ≤ 1 2 ε x for all x ∈ E and a ∈ F, and such…”

Section: Then Definementioning

confidence: 99%

“…This algebra has real rank zero and stable rank one, and the order on projections over A is determined by traces. It also has a number of other nice properties: it is exact, it satisfies the Universal Coefficient Theorem, it is approximately divisible in the sense of [3], it is a direct limit of residually finite dimensional C*-algebras, and it satisfies the local approximation property of Popa (is a Popa algebra in the sense of Definition 1.2 of [8]) and is hence quasidiagonal (by Theorem 1.2 of [30]). According to Corollary 7.21 of [8], the algebra A does not have tracial rank zero.…”

Section: Examplesmentioning

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: Then Definesupporting

confidence: 67%

“…• A is approximately divisible in the sense of [3], all quasitraces on A are traces, and projections in A distinguish the traces on A.…”

Section: The Tracial Rokhlin Propertymentioning

confidence: 99%

“…It follows from Corollary 2.10 of [3] that there is a finite dimensional unital subalgebra E ⊂ A such that [a, x] ≤ 1 2 ε x for all x ∈ E and a ∈ F, and such…”

Section: Then Definementioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

See 2 more Smart Citations

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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show abstract

“…, ∞ there exists a simple, separable and unital AH-algebra A of stable rank n. In particular, the group of invertible elements in A is not dense. And, like the examples in [20], A does not have slow dimension growth and is not approximately divisible; see [5].…”

Section: Introductionmentioning

confidence: 92%