Atomic representations of Cuntz algebras

Dutkay, Dorin Ervin; Haussermann, John; Jørgensen, Palle E. T.

doi:10.1016/j.jmaa.2014.07.005

Cited by 18 publications

(43 citation statements)

References 28 publications

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“…as claimed in (21). For the converse, suppose that the measures µ S , µ T are equivalent and there is a function h on Λ ∞ satisfying (20) and (21). Then define W :…”

Section: Disjoint and Irreducible Representationsmentioning

confidence: 99%

“…We now record some properties of P which we will rely on in the sequel. The equations below are the analogues for k-graphs of Equations (2.7) and (2.8) and (2.13) of [20]. (a) For λ, η ∈ Λ with s(λ) = r(η), we have t λ P (Z(η))t * λ = P (σ λ (Z(η))), where σ λ is the prefixing map on Λ ∞ given in Equation (4).…”

Section: Projection Valued Measuresmentioning

confidence: 99%

“…[17,18,40,27,28,26]), representations of Cuntz-Krieger algebras have been linked to fractals and Cantor sets [48,35,25,26] and to the endomorphism group of a Hilbert space [8,39]. Indeed, the astonishing goal of identifying both discrete and continuous series of representations of Cuntz (and to some extent Cuntz-Krieger) C * -algebras, was accomplished in [19,20,5], building on the pioneering results of [7].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Monic representations of finite higher-rank graphs

Farsi¹,

Gillaspy²,

Jørgensen³

et al. 2018

Ergod. Th. Dynam. Sys.

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In this paper we define the notion of monic representation for the C * -algebras of finite higher-rank graphs with no sources, and undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative C * -algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the Λ-semibranching representations previously studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal representation model for nonnegative monic representations.

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“…as claimed in (21). For the converse, suppose that the measures µ S , µ T are equivalent and there is a function h on Λ ∞ satisfying (20) and (21). Then define W :…”

Section: Disjoint and Irreducible Representationsmentioning

confidence: 99%

Section: Projection Valued Measuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Monic representations of finite higher-rank graphs

Farsi¹,

Gillaspy²,

Jørgensen³

et al. 2018

Ergod. Th. Dynam. Sys.

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“…(2.2) and then extending it by the usual Kolmogorov procedure (see [DHJ13] for details). We call this the projection-valued measure associated to the representation.…”

Section: Preliminaries: the Cuntz Algebra And Symbolic Dynamicsmentioning

confidence: 99%

Representations of Cuntz algebras associated to quasi-stationary Markov measures

Dutkay¹,

Jørgensen²

2014

Ergod. Th. Dynam. Sys.

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In this paper, we answer the question of equivalence, or singularity, of two given quasi-stationary Markov measures on one-sided infinite words, as well as the corresponding question of equivalence of associated Cuntz algebra O N -representations. We do this by associating certain monic representations of O N to quasi-stationary Markov measures and then proving that equivalence for a pair of measures is decided by unitary equivalence of the corresponding pair of representations.

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“…also [27,28,75,38,36]), representations of Cuntz-Krieger algebras have been linked to fractals and Cantor sets [90,61,35,36] and to the endomorphism group of a Hilbert space [14,72]. Indeed, the astonishing goal of identifying both discrete and continuous series of representations of Cuntz (and to some extent Cuntz-Krieger) algebras, was accomplished in [30,31,10], building on the pioneering results of [12].…”

Section: Introductionmentioning

confidence: 99%

Separable representations, KMS states, and wavelets for higher-rank graphs

Farsi

Gillaspy

Kang

et al. 2016

Journal of Mathematical Analysis and Applications

124

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Let Λ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of C * (Λ) on certain separable Hilbert spaces of the form L 2 (X, µ), by introducing the notion of a Λ-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, if Λ is aperiodic, we obtain a faithful representation ofwhere M is the Perron-Frobenius probability measure on the infinite path space Λ ∞ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a Λ-semibranching function system gives rise to KMS states for C * (Λ). For the higher-rank graphs of Robertson and Steger, we also obtain a representation of, where X is a fractal subspace of [0, 1] by embedding Λ ∞ into [0, 1] as a fractal subset X of [0,1]. In this latter case we additionally show that there exists a KMS state for C * (Λ) whose inverse temperature is equal to the Hausdorff dimension of X. Finally, we construct a wavelet system for L 2 (Λ ∞ , M ) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.2010 Mathematics Subject Classification: 46L05.

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Atomic representations of Cuntz algebras

Cited by 18 publications

References 28 publications

Monic representations of finite higher-rank graphs

Monic representations of finite higher-rank graphs

Representations of Cuntz algebras associated to quasi-stationary Markov measures

Separable representations, KMS states, and wavelets for higher-rank graphs

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