Operator algebras associated with polymorphisms

Arzumanian, V.

doi:10.1007/s10958-007-0444-3

Cited by 9 publications

(13 citation statements)

References 9 publications

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“…Along with that type of C * -algebras, the theory develops rapidly of C * -algebras generated by commutative semigroups of endomorphisms (see [7][8][9][10]). …”

Section: Introductionmentioning

confidence: 99%

C*-algebras generated by cancellative semigroups

Grigoryan

Salakhutdinov

2010

Sib Math J

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“…Along with that type of C * -algebras, the theory develops rapidly of C * -algebras generated by commutative semigroups of endomorphisms (see [7][8][9][10]). …”

Section: Introductionmentioning

confidence: 99%

C*-algebras generated by cancellative semigroups

Grigoryan

Salakhutdinov

2010

Sib Math J

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“…In the recent years, substantial efforts are made in establishing broad interplay between C * -algebras and non-invertible dynamical systems, actions of semigroups, equivalence relations, (semi-)groupoids, correspondences (see for example works by Exel [21,22], Exel and Vershik [23] Arzumanian and Vershik [4,5], Deaconu [13], Renault [39][40][41], Adji, Laca, Nielsen and Raeburn [1], an Huef and Raeburn [2], Bratteli, Jorgensen and Evans [6], Bratteli and Jorgensen [7,8], Dutkay and Jorgensen [14][15][16][17], Jorgensen [25], Dai and Larson [11], Kajiwara and Watatani [26], Kawamura [29], Watatani [48], Ostrovskyȋ and Samoȋlenko [36], Cuntz and Krieger [10], Matsumoto [35], Eilers [19], Carlsen and Silvestrov [9], and references therein).…”

Section: Theoremmentioning

confidence: 99%

On the Exel Crossed Product of Topological Covering Maps

Carlsen

Silvestrov

2008

Acta Appl Math

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For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C * -algebras C(X) α,ᏸ N introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X) α,ᏸ N is a maximal abelian C * -subalgebra of C(X) α,ᏸ N; any nontrivial two sided ideal of C(X) α,ᏸ N has non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X) α,ᏸ N is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C * -algebras of homeomorphism dynamical systems.Keywords Crossed product algebra · Covering map · Topologically free dynamical system · Maximal abelian subalgebra · Ideals Mathematics Subject Classification (2000) 47L65 · 46L55 · 47L40 · 54H20 · 37B05 · 54H15

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“…Note that if τ is continuous, by (3) and the density of I 0 in I and X 0 in X, π will be a semi-conjugacy from (X, σ) onto (I, τ ). We will call σ the local homeomorphism associated with τ .…”

Section: X) Thus For a ⊂ I τ (A) = I τ I (A ∩ [A I−1 A I ]) Andmentioning

confidence: 99%

“…, n} N , for p-fold covering maps in [12], for surjective local homeomorphisms in [1] and in [2], and for general local homeomorphisms in [59]. Similar constructions in a measure-theoretic context were introduced in [3].…”

Section: X) Thus For a ⊂ I τ (A) = I τ I (A ∩ [A I−1 A I ]) Andmentioning

confidence: 99%

C*-algebras associated with interval maps

Deaconu

Shultz

2006

Trans. Amer. Math. Soc.

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Abstract. For each piecewise monotonic map τ of [0, 1], we associate a pair of C*-algebras F τ and O τ and calculate their K-groups. The algebra F τ is an AI-algebra. We characterize when F τ and O τ are simple. In those cases, F τ has a unique trace, and O τ is purely infinite with a unique KMS state. In the case that τ is Markov, these algebras include the Cuntz-Krieger algebras O A , and the associated AF-algebras F A . Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and β-transformations. For the case of interval exchange maps and of β-transformations, the C*-algebra O τ coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively.

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Operator algebras associated with polymorphisms

Cited by 9 publications

References 9 publications

C*-algebras generated by cancellative semigroups

C*-algebras generated by cancellative semigroups

On the Exel Crossed Product of Topological Covering Maps

C*-algebras associated with interval maps

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