Furstenberg transformations on irrational rotation algebras

Osaka, Hiroyuki; Phillips, N. Christopher

doi:10.1017/s0143385706000277

Cited by 29 publications

(67 citation statements)

References 32 publications

(101 reference statements)

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“…As a consequence, combining the results in [50], we also have the following. Finally we would like to mention the Furstenberg transformations on irrational rotation algebras studied recently by H. Osaka and N. C. Phillips.…”

Section: Unital Simple Ah-algebra With No Dimension Growth and With Rmentioning

confidence: 63%

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Classification of homomorphisms and dynamical systems

Lin

2006

Trans. Amer. Math. Soc.

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Abstract. Let A be a unital simple C * -algebra, with tracial rank zero and let X be a compact metric space. Suppose that h 1 , h 2 : C(X) → A are two unital monomorphisms. We show that h 1 and h 2 are approximately unitarily equivalent if and only iffor every f ∈ C(X) and every trace τ of A. Inspired by a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let X be a compact metric space and let α, β : X → X be two minimal homeomorphisms. Using the above-mentioned result, we show that two dynamical systems are approximately conjugate in that sense if and only if a K-theoretical condition is satisfied. In the case that X is the Cantor set, this notion coincides with the strong orbit equivalence of Giordano, Putnam and Skau, and the K-theoretical condition is equivalent to saying that the associate crossed product C * -algebras are isomorphic.Another application of the above-mentioned result is given for C * -dynamical systems related to a problem of Kishimoto. Let A be a unital simple AHalgebra with no dimension growth and with real rank zero, and let α ∈ Aut(A). We prove that if α r fixes a large subgroup of K 0 (A) and has the tracial Rokhlin property, then A α Z is again a unital simple AH-algebra with no dimension growth and with real rank zero.

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Section: Unital Simple Ah-algebra With No Dimension Growth and With Rmentioning

confidence: 63%

“…The fact that (i), (ii) and (iii) are equivalent (without assuming that α r * 0 | G = id G ) is established in [50]. That (iv) ⇒ (v) is given by Theorem 2.9 in [47] (see also [44]).…”

Section: K Wherementioning

confidence: 93%

Classification of homomorphisms and dynamical systems

Lin

2006

Trans. Amer. Math. Soc.

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“…These automorphisms do not satisfy the hypotheses in [22], although in these cases we believe that the crossed products are in fact tracially AF. We treat these examples in a separate paper [26]. In that paper we also show that an automorphism of a simple unital tracially AF C*-algebra A with unique trace τ has the tracial Rokhlin property if and only if all nontrivial powers of the corresponding automorphism of the factor π τ (A)…”

Section: Introductionmentioning

confidence: 89%

“…We have not checked whether our proofs still work with this assumption. Our motivation for using the definition as stated is Theorem 2.14 of [26], which under certain conditions relates the tracial Rokhlin property to a property having the form of the Rokhlin property for automorphisms of factors of type II 1 .…”

Section: The Tracial Rokhlin Propertymentioning

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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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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“…In [17], Osaka and Phillips proved that the crossed product of a unital simple TAF-algebra with a unique tracial state by an automorphism of A with the tracial Rohlin property also has a unique tracial state. In [14], H.Lin proved that the crossed product of a unital simple TAF-algebra with a unique tracial state by an automorphism of A with the cyclic Rohlin property is also a TAFalgebra.…”

Section: Proposition 36 Any Unital Simple At-algebra Of Real Rank Zmentioning

confidence: 99%

Certain Aperiodic Automorphisms of Unital Simple Projectionless C*-Algebras

Sato

2009

Int. J. Math.

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Let G be an inductive limit of finite cyclic groups and let A be a unital simple projectionless C * -algebra with K1(A) ∼ = G and with a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in Aut(A)/ WInn(A) are conjugate, where WInn(A) means the subgroup of Aut(A) consisting of automorphisms which are inner in the tracial representation.In the second part of this paper, we consider a class of unital simple C * -algebras with a unique tracial state which contains the class of unital simple AT-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and under crossed products by actions of Z with the Rohlin property. Let A be a TAF-algebra in this class. We show that for any automorphism α of A there exists an automorphism e α of A with the Rohlin property such that e α and α are asymptotically unitarily equivalent. In its proof we use an aperiodic automorphism of the Jiang-Su algebra.

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Furstenberg transformations on irrational rotation algebras

Cited by 29 publications

References 32 publications

Classification of homomorphisms and dynamical systems

Classification of homomorphisms and dynamical systems

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

Certain Aperiodic Automorphisms of Unital Simple Projectionless C*-Algebras

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