To any finite metric space X we associate the universal Hopf C ∗-algebra H coacting on X. We prove that spaces X having at most 7 points fall into one of the following classes: (1) the coaction of H is not transitive; (2) H is the algebra of functions on the automorphism group of X; (3) X is a simplex and H corresponds to a Temperley-Lieb algebra; (4) X is a product of simplices and H corresponds to a Fuss-Catalan algebra.

Abstract. Let ℓ be a length function on a group G, and let Mℓ denote the operator of pointwise multiplication by ℓ on ℓ 2 (G). Following Connes, Mℓ can be used as a “Dirac ” operator for C ∗ r (G). It defines a Lipschitz seminorm on C ∗ r (G), which defines a metric on the state space of C ∗ r (G). We investigate whether the topology from this metric coincides with the weak- ∗ topology (our definition of a “compact quantum metric space”). We give an affirmative answer for G = Zd when ℓ is a word-length, or the restriction to Zd of a norm on Rd. This works for C ∗ r (G) twisted by a 2-cocycle, and thus for non-commutative tori. Our approach involves Connes ’ cosphere algebra, and an interesting compactification of metric spaces which is closely related to geodesic rays. The group C ∗-algebras of discrete groups provide a much-studied class of “compact non-commutative spaces ” (that is, unital C ∗-algebras). In [11] Connes showed that the “Dirac ” operator of an unbounded

Abstract. On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2-sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with “quantum metric spaces”. We show how to make these ideas precise by means of Berezin quantization using coherent states. We work in the general setting of integral coadjoint orbits for compact Lie groups. On perusing the theoretical physics literature which deals with string theory and related parts of quantum field theory, one finds in many scattered places assertions that the complex matrix algebras, Mn, converge to the two-sphere, S 2, (or to related spaces) as n goes to infinity. Here S 2 is viewed as synonymous with the algebra C(S 2) of continuous complex-valued functions on S 2 (of which S 2 is the maximal-ideal space). Approximating the sphere by matrix algebras is attractive for the following reason. In trying to carry out quantum field theory on S 2 it is natural to try to proceed by approximating S 2 by finite spaces. But “lattice ” approximations coming from choosing a finite set of points in S 2 break the very important symmetry of the action of SU(2) on S 2 (via SO(3)). But SU(2) acts naturally on the matrix algebras, in a way coherent with its action on S 2, as we will recall below. So it is natural to use them to approximate C(S 2). In this setting the matrix algebras are often referred to as “fuzzy spheres”. (See [33], [34], [17], [22], [24] and references therein.) When using the approximation of S 2 by matrix algebras, the precise sense of convergence is usually not explicitly specified in the literature. Much of the literature is at a largely algebraic level, with indications that the notion of convergence which is intended involves how structure constants and important formulas change as n grows. See, for

Abstract. We develop a matricial version of Rieffel’s Gromov-Hausdorff distance for compact quantum metric spaces within the setting of operator systems and unital C ∗-algebras. Our approach yields a metric space of “isometric ” unital complete order isomorphism classes of metrized operator systems which in many cases exhibits the same convergence properties as those in the quantum metric setting, as for example in Rieffel’s approximation of the sphere by matrix algebras using Berezin quantization. Within the metric subspace of metrized unital C ∗-algebras we establish the convergence of sequences which are Cauchy with respect to a larger Leibniz distance, and we also prove an analogue of the precompactness theorems of Gromov and Rieffel. 1.

Abstract. We introduce a new distance distoq between compact quantum metric spaces. We show that distoq is Lipschitz equivalent to Rieffel’s distance distq, and give criteria for when a parameterized family of compact quantum metric spaces is continuous with respect to distoq. As applications, we show that the continuity of a parameterized family of quantum metric spaces induced by ergodic actions of a fixed compact group is determined by the multiplicities of the actions, generalizing Rieffel’s work on noncommutative tori and integral coadjoint orbits of semisimple compact connected Lie groups; we also show that the θ-deformations of Connes and Landi are continuous in the parameter θ. 1.

Abstract. We introduce an analogue for Lip-normed operator systems of the second author’s order-unit quantum Gromov-Hausdorff distance and prove that it is equal to the first author’s complete distance. This enables us to consolidate the basic theory of what might be called operator Gromov-Hausdorff convergence. In particular we establish a completeness theorem and deduce continuity in quantum tori, Berezin-Toeplitz quantizations, and θ-deformations from work of the second author. We show that approximability by Lip-normed matrix algebras is equivalent to 1-exactness of the underlying operator space and, by applying a result of Junge and Pisier, that for n ≥ 7 the set of isometry classes of n-dimensional Lip-normed operator systems is nonseparable. We also treat the question of generic complete order structure. 1.

We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation to the algebra of transverse differential operators introduced by Connes and Moscovici in order to compute a local index formula in cyclic cohomology, and to the several Hopf algebras defined by Connes and Kreimer to simplify the combinatorics of perturbative renormalization. We explain how characteristic classes for a Hopf module algebra can be obtained from the cyclic cohomology of the Hopf algebra which acts on it. Finally, we discuss the theory of noncommutative spherical manifolds and show how they arise as homogeneous spaces of certain compact quantum groups.

Abstract. Let ℓ be a length function on a group G, and let Mℓ denote the operator of pointwise multiplication by ℓ on ℓ 2 (G). Following Connes, Mℓ can be used as a “Dirac ” operator for C ∗ r (G). It defines a Lipschitz seminorm on C ∗ r (G), which defines a metric on the state space of C ∗ r(G). We show that if G is a hyperbolic group and if ℓ is a word-length function on G, then the topology from this metric coincides with the weak- ∗ topology (our definition of a “compact quantum metric space”). We show that a convenient framework is that of filtered C∗-algebras which satisfy a suitable “ Haagerup-type ” condition. We also use this framework to prove an analogous fact for certain reduced free products of C∗-algebras. The group C ∗-algebras of discrete groups provide a much-studied class of “compact non-commutative spaces ” (that is, unital C ∗-algebras). In [3] Connes showed that the “Dirac ” operator of a spectral triple (i.e.

Abstract. An AF C*-algebra has a natural filtration as an increasing sequence of finite dimensional C*-algebras. We show that it is possible to construct a Dirac operator which relates to this filtration in a natural way and which will induce a metric for the weak*-topology on the state space of the algebra. In the particular case of a UHF C*algebra the construction can be made in a way, which relates directly to the dimensions of the increasing sequence of subalgebras. The algebra of continuous functions on the Cantor set is an approximately finite dimensional C*-algebra and our investigations show, when applied to this algebra, that the proposed Dirac operators have good classical interpretations and lead to an, apparently, new way of constructing a representative for a Cantor set of any given Hausdorff dimension. At the end of the paper we study the finite dimensional full matrix algebras over the complex numbers, Mn, and show that the operation of transposition on matrices yields a spectral triple which has the property that it’s metric on the state space is exactly the norm distance. This result is then generalized to arbitrary unital C*-algebras. 1.

Let G⎛be the Heisenberg ⎞ group. 1 x z