Wavelets and spectral triples for fractal
                    representations of Cuntz algebras

Farsi, Carla; Gillaspy, Elizabeth; Julien, Antoine; Kang, Sooran; Packer, Judith

doi:10.1090/conm/687/13795

Cited by 7 publications

(33 citation statements)

References 19 publications

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“…[31]) that the maximal abelian subalgebra of B(L 2 (Λ ∞ , µ)), for any finite Borel measure µ, is the sub-algebra L ∞ (Λ ∞ , µ) consisting of multiplication operators now implies that T must be a multiplication operator too. In other words, there exists a function F µ in L ∞ (Λ ∞ , µ) such that T (f dµ) = F µ f dµ (25) for all f ∈ L 2 (Λ ∞ , µ), establishing (iii ). It remains to check the properties of the functions F µ .…”

Section: A Universal Representation For Non-negative λ-Projective Sysmentioning

confidence: 82%

“…[11,12,23,38]), k-graph C * -algebras share many of the important properties of Cuntz and Cuntz-Krieger C * -algebras, including Cuntz-Krieger uniqueness theorems and realizations as groupoid C * -algebras. Moreover, the C * -algebras of higher-rank graphs are closely linked with orbit equivalence for shift spaces [10] and with symbolic dynamics more generally [43,47,44], as well as with fractals and self-similar structures [25,26]. More links between higher-rank graphs and symbolic dynamics can be seen via [3,4] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

“…In addition to connections with wavelets (cf. [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%

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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: A Universal Representation For Non-negative λ-Projective Sysmentioning

confidence: 82%

Section: Introductionmentioning

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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“…In addition to their relevance for C * -algebraic classification [78,86], the C * -algebras of higher-rank graphs are closely linked with orbit equivalence for shift spaces [17] and with symbolic dynamics more generally [79,88,80], with fractals and self-similar structures [35,36], and with renormalization problems in physics [40]. More links between higher-rank graphs and symbolic dynamics can be seen via [8,9,7] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

“…In addition to the connections with wavelets which were indicated above (cf. 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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Spectral triples and wavelets for higher-rank graphs

Farsi

Gillaspy

Julien

et al. 2020

Journal of Mathematical Analysis and Applications

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In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph Λ, via the infinite path space Λ ∞ of Λ. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of Λ ∞ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary -Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph Λ. We then study the zeta function, abscissa of convergence, and Dixmier trace associated to the Pearson-Bellissard spectral triples of these Cantor sets, and show these spectral triples are -regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure , and show that is a rescaled version of the measure on Λ ∞ which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrami operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of 2 (Λ ∞ , ) which was constructed by Farsi et al. 2.5 below. The space of infinite paths  of a stationary -Bratteli diagram  is often a Cantor set, enabling us to study its associated Pearson-Bellissard spectral triple. Indeed, if the matrices 1 , … , are the adjacency matrices for a -graph Λ, then the space of infinite paths in Λ is homeomorphic to the Cantor set  (also called ). In other words, the Pearson-Bellissard spectral triples for stationary -Bratteli diagrams can also be viewed as spectral triples for higher-rank graphs.We then proceed to study, in Section 3, the geometrical information encoded by these spectral triples. Theorem 3.14 establishes that the Pearson-Bellissard spectral triple associated to (  Λ , ) is finitely summable, with dimension ∈ (0, 1). Section 3.3 focuses on the Dixmier traces of the spectral triples, and establishes both an integral formula for the Dixmier trace (Theorems 3.23 and 3.28) and a concrete expression for the measure induced by the Dixmier trace (Theorem 3.26). These computations also reveal that the ultrametric Cantor sets (  Λ , ) are -regular in the sense of [59, Definition 11]. Other settings in the literature in which spectral triples on Cantor sets admit an integral formula for the Dixmier trace include [13,47,17,14].In full generality, Dixmier traces are defined on the Dixmier-Macaev (also called Lorentz) ideal  1,∞ ⊆ () inside the compact operators and are computed using a generalized limit (roughly speaking, a linear functional that lies between lim sup and lim inf). Although the theory of Dixmier traces can be quite intricate, many of the computations simplify substantially in our setting, and so our treatment of the general theory will be brief; we refer the interested reader to the extensive literature on Dixmier traces and other singular traces (cf. [19, 55, 54, 1...

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Wavelets and spectral triples for fractal representations of Cuntz algebras

Cited by 7 publications

References 19 publications

Monic representations of finite higher-rank graphs

Monic representations of finite higher-rank graphs

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

Spectral triples and wavelets for higher-rank graphs

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