The Podleś sphere as a spectral metric space

Abstract

We study the spectral metric aspects of the standard Podleś sphere, which is a homogeneous space for quantum SU (2). The point of departure is the real equivariant spectral triple investigated by Da ֒ browski and Sitarz. The Dirac operator of this spectral triple interprets the standard Podleś sphere as a 0-dimensional space and is therefore not isospectral to the Dirac operator on the 2-sphere. We show that the seminorm coming from commutators with this Dirac operator provides the Podleś sphere with the structure of a compact quantum metric space in the sense of Rieffel.2010 Mathematics Subject Classification. 58B32,58B34;46L89,46L30.

Polynomial approximation of quantum Lipschitz functions

“…The main focus in the present paper is the Podleś sphere S 2 q [22], which forms the base of a spectral triple for the Dąbrowski-Sitarz Dirac operator D q [9], whose associated seminorm L max Dq turns C(S 2 q ) into a compact quantum metric space, as proven in [2]. However, the natural point of departure when studying S 2 q is actually the associated coordinate algebra O(S 2 q ) which is a subalgebra of the Lipschitz algebra C Lip (S 2 q ) := Dom(L max Dq ), and one may therefore restrict L max Dq to O(S 2 q ) to obtain another Lip-norm L Dq .…”

Inductive Limits of C*-Algebras and Compact Quantum Metric Spaces

“…But to discuss convergence of C*-algebras, one must first equip the C*-algebras with the quantum metrics. There are many C*-algebras that have been equipped with quantum metrics, including certain group C*-algebras [10,32,35], quantum tori and fuzzy tori [20,33], noncommutative solenoids [29], the quantum Podleś sphere with D ąbrowski-Sitarz spectral triple [3,13], and many more including unital approximately finite-dimensional (AF) algebras [1,6]. Now, in the case of unital AF algebras equipped with faithful tracial states, in [4], quantum metrics were placed in such a way that any inductive sequence that formed the given unital AF algebra converges to the AF algebra in the metric sense of propinquity so that the authors were able to also establish convergence of classes of AF algebras such as uniformly hyperfinite (UHF) algebras [19] and Effros-Shen algebras [15].…”

“…Thus, in this paper, we accomplish this task under suitable sufficient conditions to obtain our new quantum metrics on AF algebras. The main fact we use is that the dual Gromov-Hausdorff propinquity [21] is complete on certain classes of compact quantum metric spaces, and thus bestows a method for forming limits of [3] quantum metric spaces. Of course, this limit may not be an inductive limit of quantum metric spaces in any categorical sense, but the idea is to combine the categorical notion of inductive limit of C*-algebras with the metric limit formed by completeness with respect to propinquity.…”

Heisenberg Modules over Quantum 2-tori are Metrized Quantum Vector Bundles

“…Noncommutative metric geometry [6,28,30] studies noncommutative generalizations of Lipschitz algebras, defined as follows. Leibniz quantum compact metric spaces, and more generally quasi-Leibniz quantum compact metric spaces (a generalization we will not need in this paper), form a category with the appropriate notion of Lipschitz morphisms [21], containing such important examples as quantum tori [28], Connes-Landi spheres [22], group C*-algebras for Hyperbolic groups and nilpotent groups [29,23], AF algebras [12], Podlès spheres [2], certain C*-crossed-products [1], among others. Any compact metric space (X, d) give rise to the Leibniz quantum compact metric space (C(X), Lip) where C(X) is the C*-algebra of C-valued continuous functions over X, and Lip is the Lipschitz seminorm induced by d.…”