Classification of the<i>C</i><sup>∗</sup>-algebras associated with minimal rotations

Riedel, Norbert

doi:10.2140/pjm.1982.101.153

Cited by 20 publications

(16 citation statements)

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“…which can be obtained by representing the crossed products faithfully on L 2 ( Z p × T) and L 2 (Z×Z p ) via the canonical regular representations, respectively, and then check that conjugation with the Plancherel isomorphism Ψ : L 2 ( Z p × T) → L 2 (Z p × Z) ∼ = L 2 (Z × Z p ) induces the desired isomorphism. Thus we can apply [28,Theorem 3.6] which implies that K 0 (C 0 (Z p ) ⋊ Z) is isomorphic to the group Z[ 1 p ] = { k p l : k ∈ Z, l ∈ N 0 } (note that the crossed product in question is also isomorphic to the well known Bunce-Deddens algebra). The abelian group Z[ 1 p ] is not isomorphic to any direct sum of copies of Z.…”

Section: K-theory Of Crossed Products By Actions On Totally Disconnecmentioning

confidence: 99%

On the K-Theory of Crossed Products by Automorphic Semigroup Actions

Cuntz¹,

Echterhoff²,

Li³

2013

The Quarterly Journal of Mathematics

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Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies the Toeplitz condition of [24] and that the Baum-Connes conjecture holds for G. We prove a formula describing the Ktheory of the reduced crossed product A ⋊α,r P by any automorphic action of P . This formula is obtained as a consequence of a result on the K-theory of crossed products for special actions of G on totally disconnected spaces. We apply our result to various examples including left Ore semigroups and quasilattice ordered semigroups. We also use the results to show that for certain semigroups P , including the ax + b-semigroup R ⋊ R × for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras C * λ (P ) and C * ρ (P ) coincide, although the structure of these algebras can be very different. Preliminaries on totally disconnected spacesRecall that a locally compact Hausdorff space Ω is totally disconnected if and only if its topology has a basis of compact open subsets. The corresponding algebras C 0 (Ω) of continuous functions which vanish at infinity are precisely the commutative AF-Algebras. In what follows, if V ⊆ Ω, then 1 V : Ω → C denotes the characteristic function of V . Definition 2.1. Let Ω be a totally disconnected locally compact Hausdorff space and let V be a family of compact open subsets in Ω. Moreover, let U c (Ω) denote the set of all compact open subsets of Ω. Then we say that V is a generating family of the compact open sets of Ω if U c (Ω) coincides with the smallest family U of compact open sets in Ω which contains V and which is closed under finite intersections, finite unions, and under taking differences U W with U, W ∈ U . Lemma 2.2. Suppose that V is a family of compact open sets in the totally disconnected space Ω. Then the following are equivalent (1) The set {1 V : V ∈ V} generates C 0 (Ω) as a C*-algebra. (2) The set V generates U c (Ω) in the sense of Definition 2.1. Moreover, if V is closed under taking finite intersections, then (1) and (2) are equivalent to(3) span{1 V : V ∈ V} is a dense subalgebra of C 0 (Ω) containing span{1 U : U ∈ U c (Ω)}.

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Section: K-theory Of Crossed Products By Actions On Totally Disconnecmentioning

confidence: 99%

On the K-Theory of Crossed Products by Automorphic Semigroup Actions

Cuntz¹,

Echterhoff²,

Li³

2013

The Quarterly Journal of Mathematics

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“…Moreover, both can be realized, in the first case using results in [21] and in the second case it is realized by a Sturmian subshift. Relations between additive eigenvalues and topological invariants can be found in [27,26,23,11], but they do not apply to Cantor systems.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues and strong orbit equivalence

Cortéz

Durand²,

Petite³

2015

Ergod. Th. Dynam. Sys.

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Abstract. We give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues E(X, T ) of the minimal Cantor system (X, T ) is a subgroup of the intersection I(X, T ) of all the images of the dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated to (X, T ) is trivial, the quotient group I(X, T )/E(X, T ) is torsion free. We give examples with non trivial infinitesimal subgroups where this property fails. We also provide some realization results.

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“…D{φ) = I 2 , have been classified by Riedel [13,Corollary 3.7]. F(φ(x, y)) -F(x, y) = c, for all (x, y) € T 2 , where c is a real constant.…”

Section: Remark the Quantity Det(d(φ) -I2mentioning

confidence: 99%

Quasi-rotationC^∗-algebras

Rouhani¹

1991

Pacific J. Math.

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The main result in this paper is to classify the isomorphism classes of certain non-commutative 3-tori obtained by taking the crossed product C*-algebra of continuous functions on the 2-torus T 2 by the irrational affine quasi-rotations. Each such quasi-rotation is represented by a pair (a, A), where a e Ύ 2 and A e GL(2, Z), and its associated C* -algebra is shown to be determined (up to isomorphism) by an analogue of the rotation angle, namely its primitive eigenvalue, by its orientation άel(A) = ±1 and a certain positive integer m(A) which comes from the K\ -group of the algebra and which determines the conjugacy class of A in GL(2, Z).

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Classification of theC^∗-algebras associated with minimal rotations

Cited by 20 publications

References 5 publications

On the K-Theory of Crossed Products by Automorphic Semigroup Actions

On the K-Theory of Crossed Products by Automorphic Semigroup Actions

Eigenvalues and strong orbit equivalence

Quasi-rotationC^∗-algebras

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Classification of theC∗-algebras associated with minimal rotations

Cited by 20 publications

References 5 publications

On the K-Theory of Crossed Products by Automorphic Semigroup Actions

On the K-Theory of Crossed Products by Automorphic Semigroup Actions

Eigenvalues and strong orbit equivalence

Quasi-rotationC∗-algebras

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Product

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Classification of theC^∗-algebras associated with minimal rotations

Quasi-rotationC^∗-algebras