We introduce the quantum Gromov-Hausdorff propinquity, a new distance between quantum compact metric spaces, which extends the Gromov-Hausdorff distance to noncommutative geometry and strengthens Rieffel's quantum Gromov-Hausdorff distance and Rieffel's proximity by making *-isomorphism a necessary condition for distance zero, while being well adapted to Leibniz seminorms. This work offers a natural solution to the long-standing problem of finding a framework for the development of a theory of Leibniz Lip-norms over C*algebras.

Abstract. We construct quantum metric structures on unital AF algebras with a faithful tracial state, and prove that for such metrics, AF algebras are limits of their defining inductive sequences of finite dimensional C*-algebras for the quantum propinquity. We then study the geometry, for the quantum propinquity, of three natural classes of AF algebras equipped with our quantum metrics: the UHF algebras, the Effros-Shen AF algebras associated with continued fraction expansions of irrationals, and the Cantor space, on which our construction recovers traditional ultrametrics. We also exhibit several compact classes of AF algebras for the quantum propinquity and show continuity of our family of Lip-norms on a fixed AF algebra. Our work thus brings AF algebras into the realm of noncommutative metric geometry.

SUMMARYWe generated a knockout mouse for the neuronalspecific β-tubulin isoform Tubb3 to investigate its role in nervous system formation and maintenance. Tubb3−/− mice have no detectable neurobehavioral or neuropathological deficits, and upregulation of mRNA and protein of the remaining β-tubulin isotypes results in equivalent total b-tubulin levels in Tubb3−/− and wild-type mice. Despite similar levels of total β-tubulin, adult dorsal root ganglia lacking TUBB3 have decreased growth cone microtubule dynamics and a decreased neurite outgrowth rate of 22% in vitro and in vivo. The effect of the 22% slower growth rate is exacerbated for sensory recovery, where fibers must reinnervate the full volume of the skin to recover touch function. Overall, these data reveal that, while TUBB3 is not required for formation of the nervous system, it has a specific role in the rate of peripheral axon regeneration that cannot be replaced by other β-tubulins.

We establish that, given a compact Abelian group G endowed with a continuous length function l and a sequence (H n ) n∈N of closed subgroups of G converging to G for the Hausdorff distance induced by l, then C * G, is the quantum Gromov-Hausdorff limit of any sequence C * H n , n n∈N for the natural quantum metric structures and when the lifts of n to G converge pointwise to . This allows us in particular to approximate the quantum tori by finite-dimensional C * -algebras for the quantum Gromov-Hausdorff distance. Moreover, we also establish that if the length function l is allowed to vary, we can collapse quantum metric spaces to various quotient quantum metric spaces.

We prove that noncommutative solenoids are limits, in the sense of the Gromov-Hausdorff propinquity, of quantum tori. From this observation, we prove that noncommutative solenoids can be approximated by finite dimensional quantum compact metric spaces, and that they form a continuous family of quantum compact metric spaces over the space of multipliers of the solenoid, properly metrized. IntroductionThe quantum Gromov-Hausdorff propinquity, introduced by the first author [16,13], is a distance on quantum compact metric spaces which extends the topology of the Gromov-Hausdorff distance [7,6]. Quantum metric spaces are generalizations of Lipschitz algebras [28] first discussed by Connes [3] and formalized by Rieffel [22]. The propinquity strengthens Rieffel's quantum Gromov-Hausdorff distance [26] to be well-adapted to the C*-algebraic framework, in particular by making *-isomorphism a necessary condition for distance zero [14]. The propinquity thus allows us to address questions from mathematical physics, such as the problem of finite dimensional approximations of quantum space times [4,20,5,27], [21, Ch. 7]. Matricial approximations of physical theory motivates our project, which requires, at this early stage, the study of many different examples of quantum spaces.Recently, the first author proved that quantum tori form a continuous family for the propinquity, and admit finite dimensional approximations via so-called fuzzy tori [10]. This paper, together with the work on AF algebras done in [1], explores the connection between our geometric approach to limits of C*-algebras and the now well studied approach via inductive limits, which itself played a role is quantum statistical mechanics [2]. We thus bring noncommutative solenoids, studied by the authors in [17,18,19], and which are inductive limits of quantum tori, into the realm of noncommutative metric geometry. Our techniques apply to more general inductive limits on which projective limits of compact metrizable groups act ergodically. Noncommutative solenoids are interesting examples since they also are C*-crossed products, whose metric structures are still a challenge to understand. Irrational noncommutative solenoids [17] are non-type I C*-algebras, and many Date