Crossed product extensions of spectral triples

Iochum, Bruno; Masson, Thierry

doi:10.4171/jncg/229

Cited by 16 publications

(15 citation statements)

References 28 publications

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“…The Euclidean version of κ-Minkowski space has also been studied from the perspective of spectral triples [85][86][87] via the star-product realizations [88][89][90]. In this framework the algebra (35) is faithfully represented on the Hilbert space L 2 ðR nþ1 Þ through a left regular group representation.…”

Section: κ-Minkowski Spacetimementioning

confidence: 99%

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Spectral dimensions and dimension spectra of quantum spacetimes

Eckstein

Trześniewski

2020

Phys. Rev. D

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Different approaches to quantum gravity generally predict that the dimension of spacetime at the fundamental level is not 4. The principal tool to measure how the dimension changes between the IR and UV scales of the theory is the spectral dimension. On the other hand, the noncommutative-geometric perspective suggests that quantum spacetimes ought to be characterized by a discrete complex set-the dimension spectrum. We show that these two notions complement each other and the dimension spectrum is very useful in unraveling the UV behavior of the spectral dimension. We perform an extended analysis highlighting the trouble spots and illustrate the general results with two concrete examples: the quantum sphere and the κ-Minkowski spacetime, for a few different Laplacians. In particular, we find that the spectral dimensions of the former exhibit log-periodic oscillations, the amplitude of which decays rapidly as the deformation parameter tends to the classical value. In contrast, no such oscillations occur for either of the three considered Laplacians on the κ-Minkowski spacetime.

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Section: κ-Minkowski Spacetimementioning

confidence: 99%

“…[86,89]). Fortunately, it turns out that the regularizing function factors out in trace formulas and contributes just a multiplicative factor kfk L 2 [86,87]. This means that the return probability does not depend on the choice of the IR regularization and can be computed via the heat kernel formula (6).…”

Section: κ-Minkowski Spacetimementioning

confidence: 99%

Spectral dimensions and dimension spectra of quantum spacetimes

Eckstein

Trześniewski

2020

Phys. Rev. D

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“…The reader may, however, consult the paper [KRS12] where a similar structure is encountered. For other interesting approaches to the non-commutative geometry of crossed products (and many other things) we refer to [PoWa18,IoMa16,GMR19] and of course [CoMo95,CoMo08,Mos10]. These papers are not essential for the understanding of the present text, but the references are included to put the present paper into a broader context.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of compact quantum metric spaces

Kaad¹,

Kyed²

2020

Ergod. Th. Dynam. Sys.

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We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. Moreover, we provide a flexible set of assumptions ensuring that a continuous family of $\ast$ -automorphisms of a compact quantum metric space yields a field of crossed product algebras which varies continuously in Rieffel’s quantum Gromov–Hausdorff distance. Finally, we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney $C^{1}$ -topology.

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“…Remark 2.7. We refer to [38,51,39,57,68,79] for the constructions of various other examples of twisted spectral triples.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative geometry and conformal geometry. I. Local index formula and conformal invariants

Ponge

Wang

2018

J. Noncommut. Geom.

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This paper is part of a series of articles on noncommutative geometry and conformal geometry. In this paper, we reformulate the local index formula in conformal geometry in such a way to take into account of the action of conformal diffeomorphisms. We also construct and compute a whole new family of geometric conformal invariants associated with conformal diffeomorphisms. This includes conformal invariants associated with equivariant characteristic classes. The approach of this paper involves using various tools from noncommutative geometry, such as twisted spectral triples and cyclic theory. An important step is to establish the conformal invariance of the Connes-Chern character of the conformal Dirac spectral triple of Connes-Moscovici. Ultimately, however, the main results of the paper are stated in a purely differentialgeometric fashion.2. Index Theory for Twisted Spectral Triples.

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Crossed product extensions of spectral triples

Cited by 16 publications

References 28 publications

Spectral dimensions and dimension spectra of quantum spacetimes

Spectral dimensions and dimension spectra of quantum spacetimes

Dynamics of compact quantum metric spaces

Noncommutative geometry and conformal geometry. I. Local index formula and conformal invariants

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