Monic representations of the Cuntz algebra and Markov measures

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

doi:10.1016/j.jfa.2014.05.016

Cited by 26 publications

(90 citation statements)

References 22 publications

(15 reference statements)

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“…The second technical tool which underpins our analysis of the monic representations of C * (Λ) is the projection valued measure associated to a representation of C * (Λ). Our work in this section is inspired by Dutkay, Haussermann, and Jorgensen [19,20].…”

Section: Projection Valued Measuresmentioning

confidence: 99%

“…Since W intertwines the representations {T λ } λ∈Λ , {T ′ λ } λ∈Λ of C * (Λ), we must have W = 0; hence h = 0, so µ, µ ′ are mutually singular. In fact, these Markov measures are mutually singular with the Perron-Frobenius measure [19], [33]. From the projection valued measure P associated to {t λ : λ ∈ Λ} as in Theorem 3.9, we obtain a representation π : C(Λ ∞ ) → B(H):…”

Section: Disjoint and Irreducible Representationsmentioning

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%

“…In this paper, we refine the Λ-semibranching function systems into a crucial technical tool for studying monic representations, namely the Λ-projective systems of Definition 3.1. Monic representations also have strong connections to Markov measures [19,5] and Nelson's universal representation of an abelian algebra [42]. Indeed, studying monic representations enables us to convert questions about the representation theory of higher-rank graphs into measure-theoretic questions (see Theorems 3.11 and 3.13 below).…”

Section: Introductionmentioning

confidence: 99%

See 3 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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Section: Projection Valued Measuresmentioning

confidence: 99%

Section: Disjoint and Irreducible Representationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…[DJ14] Let (S i ) i∈Z N be a representation of O N . The representation is monic if and only if it is unitarily equivalent to a representation associated to a monic system.…”

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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Wavelets and Graph C ∗-Algebras

Farsi

Gillaspy

Kang

et al. 2017

Excursions in Harmonic Analysis, Volume 5

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Here we give an overview on the connection between wavelet theory and representation theory for graph C * -algebras, including the higher-rank graph C * -algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In [20], we introduced the "cubical wavelets" associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of [22] to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs.2010 Mathematics Subject Classification: 46L05, 42C40

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Monic representations of the Cuntz algebra and Markov measures

Cited by 26 publications

References 22 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

Wavelets and Graph C ∗-Algebras

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