Contractive Spectral Triples for Crossed Products

Paterson, Alan L. T.

doi:10.7146/math.scand.a-17112

Cited by 17 publications

(13 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the case where (B, H, D) is a unital spectral triple and the * -automorphism β : B → B is equicontinuous, we may choose λ = µ = 1 so that = id : 2 (Z) ⊗H → 2 (Z) ⊗H . The seminorm L = L id then coincides exactly with the seminorm investigated in [BMR10,§3] and can be seen to arise from a unital spectral triple over the reduced crossed product C * r (Z, B); see also [HSWZ13,Pat14]. In the more general setting where the * -automorphism β : B → B is only quasi-isometric we still have the seminorm L : A → [0, ∞) and we shall relate it to metrics on the state space of C * r (Z, B) in §5.…”

Section: It Thus Suffices To Show That the Diagonal Operatormentioning

confidence: 52%

Dynamics of compact quantum metric spaces

Kaad¹,

Kyed²

2020

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

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.

show abstract

Section: It Thus Suffices To Show That the Diagonal Operatormentioning

confidence: 52%

Dynamics of compact quantum metric spaces

Kaad¹,

Kyed²

2020

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

“…which is clearly a bounded function of ω for any a ∈ A 1 . Moreover, such a ∈ A 1 sends the core C of our selfadjoint operator D u to itself and following [48,Proposition A.1,p. 293], this suffices to ensure that a ∈ A 1 sends the domain of D u to itself.…”

Section: Conformally Twisted Spectral Triplesmentioning

confidence: 99%

On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C<sup>*</sup>-Dynamical Systems

Fathizadeh

Gabriel

2016

SIGMA

View full text Add to dashboard Cite

Abstract. The analog of the Chern-Gauss-Bonnet theorem is studied for a C * -dynamical system consisting of a C * -algebra A equipped with an ergodic action of a compact Lie group G. The structure of the Lie algebra g of G is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra A ⊂ A as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique G-invariant state on A, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on A and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.

show abstract

“…This bundle is noncompact in general, hence the need to work with non-unital Lipschitz pairs -and invoke our results described in this section. A follow-up of [8] using our bounded-Lipschitz metric can be found in [66].…”

Section: 31mentioning

confidence: 99%

Quantum Metric Spaces and the Gromov-Hausdorff Propinquity

Latrémolière¹

2016

Noncommutative Geometry and Optimal Transport

View full text Add to dashboard Cite

We present a survey of the dual Gromov-Hausdorff propinquity, a noncommutative analogue of the Gromov-Hausdorff distance which we introduced to provide a framework for the study of the noncommutative metric properties of C*-algebras. We first review the notions of quantum locally compact metric spaces, and present various examples of such structures. We then explain the construction of the dual Gromov-Hausdorff propinquity, first in the context of quasi-Leibniz quantum compact metric spaces, and then in the context of pointed proper quantum metric spaces. We include a few new result concerning perturbations of the metrics on Leibniz quantum compact metric spaces in relation with the dual Gromov-Hausdorff propinquity.

show abstract

Contractive Spectral Triples for Crossed Products

Cited by 17 publications

References 19 publications

Dynamics of compact quantum metric spaces

Dynamics of compact quantum metric spaces

On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C<sup>*</sup>-Dynamical Systems

Quantum Metric Spaces and the Gromov-Hausdorff Propinquity

Contact Info

Product

Resources

About