Dynamics of compact quantum metric spaces

Abstract

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.

The quantum metric structure of quantum SU(2)

“…We remark that the Monge-Kantorovič metric d L is not, strictly speaking, a metric since it can, a priori, take the value infinity. In fact, it can be proved that if ker(L) contains non-scalar elements, then there exist states µ 0 and ν 0 on X such that d L (µ 0 , ν 0 ) = ∞; see for example [KK21,Lemma 2.2]. This possibility is excluded when (X, L) is a compact quantum metric space in the following sense: Definition 2.3.…”

“…The existence of such a distance function allows one to study the class of compact quantum metric spaces from a more analytical point of view, and opens the possibility to investigate a wealth of natural continuity questions. Over the past two decades, many positive answers have been obtained, and examples include Rieffel's fundamental result that the 2-sphere can be approximated by the fuzzy spheres (matrix algebras) [Rie04b], as well as the more recent proof [AKK21a] that the Podleś spheres S 2 q vary continuously in the deformation parameter q ∈ (0, 1]; for many more examples see [Agu17,KK21,Lat05,LP18,Li09,Rie04a].…”

An Ideal Convergence

“…As AF-algebras are tightly connected with problems from dynamics, specifically with minimal actions on Cantor sets, the connection between the metric properties of AF-algebras and their actions with a metric picture of dynamical system could prove very interesting. The interplay between dynamics, metric geometry, and compact quantum metric spaces is a subject for current and future investigation (for some important recent work in this direction see [KK21]). Spectral triples, as a means to define a noncommutative Riemannian structure, are tools of central importance in the study of noncommutative geometry, and their metric properties are yet to be fully understood, though hopefully future work based upon the subject we brushed upon in this article will prove enlightening.…”

Polynomial approximation of quantum Lipschitz functions

“…This allows one to study the class of compact quantum metric spaces from an analytical point of view, and ask questions pertaining to continuity and convergence of families of compact quantum metric spaces, see e.g. [1,14,19,21,29,30] for examples of this. The main focus in the present paper is the Podleś sphere S 2 q [26], which forms the base of a spectral triple for the Dąbrowski-Sitarz Dirac operator D q [10], whose associated seminorm L max Dq turns C(S 2 q ) into a compact quantum metric space, as proven in [2].…”