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43
Metrics on state spaces
 Doc. Math
, 1999
"... This article is dedicated to Richard V. Kadison in anticipation of his completing his seventyfifth circumnavigation of the sun. Abstract. In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly noncommu ..."
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Cited by 37 (4 self)
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This article is dedicated to Richard V. Kadison in anticipation of his completing his seventyfifth circumnavigation of the sun. Abstract. In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly noncommutative compact spaces are usually not determined by the restriction of the metric they define on the state space, to the extreme points of the state space. We characterize the Lipschitz norms which are determined by their metric on the whole state space as being those which are lower semicontinuous. We show that their domain of Lipschitz elements can be enlarged so as to form a dual Banach space, which generalizes the situation for ordinary Lipschitz seminorms. We give a characterization of the metrics on state spaces which come from Lipschitz seminorms. The natural (broader) setting for these results is provided by the “function spaces” of Kadison. A variety of methods for constructing Lipschitz seminorms is indicated. In noncommutative geometry (based on C ∗algebras), the natural way to specify a metric is by means of a suitable “Lipschitz seminorm”. This idea was first suggested by Connes [C1] and developed further in [C2, C3]. Connes pointed out [C1, C2] that from a Lipschitz seminorm one obtains in a simple way an ordinary metric on the state space of the C ∗algebra. This metric generalizes the Monge–Kantorovich metric on probability measures [KA, Ra, RR]. In this article we make more precise the relationship between metrics on the state space and Lipschitz seminorms. Let ρ be an ordinary metric on a compact space X. The Lipschitz seminorm, Lρ, determined by ρ is defined on functions f on X by (0.1) Lρ(f) = sup{f(x) − f(y)/ρ(x, y) : x ̸ = y}.
GromovHausdorff distance for quantum metric spaces
 Mem. Amer. Math. Soc
"... Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distanc ..."
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Cited by 35 (5 self)
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Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distance. We show that the basic theorems of the classical theory have natural quantum analogues. Our main example involves the quantum tori, Aθ. We show, for consistently defined “metrics”, that if a sequence {θn} of parameters converges to a parameter θ, then the sequence {Aθn} of quantum tori converges in quantum Gromov–Hausdorff distance to Aθ. 1.
Group C ∗ algebras as compact quantum metric spaces
 Doc. Math
"... 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). ..."
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Cited by 23 (0 self)
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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 wordlength, or the restriction to Zd of a norm on Rd. This works for C ∗ r (G) twisted by a 2cocycle, and thus for noncommutative 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 muchstudied class of “compact noncommutative spaces ” (that is, unital C ∗algebras). In [11] Connes showed that the “Dirac ” operator of an unbounded
Matrix algebras converge to the sphere for quantum GromovHausdorff distance
 Mem. Amer. Math. Soc
"... Abstract. On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with “quantum metric spaces”. We show how t ..."
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Cited by 20 (3 self)
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Abstract. On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2sphere (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 twosphere, 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 complexvalued functions on S 2 (of which S 2 is the maximalideal 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
KMS states, entropy and the variational principle in full C∗dynamical systems
 COMM. MATH. PHYS
, 1999
"... To any periodic and full C ∗ –dynamical system (A, α, R) a certain invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron–Frobenius theorem asserts the existence of KMS ..."
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Cited by 19 (6 self)
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To any periodic and full C ∗ –dynamical system (A, α, R) a certain invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron–Frobenius theorem asserts the existence of KMS states at inverse temperatures the logarithm of the inner and outer spectral radii of s. Such KMS states are called extremal. Examples arising from subshifts in symbolic dynamics, self–similar sets in fractal geometry and noncommutative metric spaces are discussed. Certain subshifts are naturally associated to the system, and criteria for the equality of their topological entropy and inverse temperatures of extremal KMS states are given. Also, unital completely positive maps σ {xj} implemented by partitions of unity {xj} of grade 1 are considered, resembling the ‘canonical endomorphism’ of the Cuntz algebras. The relationship between the Voiculescu topological entropy of σ {xj} and the topological entropy of the associated subshift is studied. Similarly, the measure–theoretic entropy of σ {xj}, in the sense of Connes–Narnhofer–Thirring, is compared to the classical measure–theoretic entropy of the subshift. A noncommutative analogue of the classical variational principle for the entropy of subshifts is obtained for those maps σ {xj} for which {xj} generates a Matsumoto C ∗ –algebra. When {xj} generates a Cuntz–Krieger algebra, an explicit construction of states with maximal entropy from KMS states at maximal inverse temperatures is done.
Matricial quantum GromovHausdorff distance
 J. Funct. Anal
"... Abstract. We develop a matricial version of Rieffel’s GromovHausdorff 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 ..."
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Cited by 18 (1 self)
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Abstract. We develop a matricial version of Rieffel’s GromovHausdorff 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.
Distance in finite spaces from non commutative geometry
 Journ. Geom. Phys
"... Following the general principles of noncommutative geometry, it is possible to define a metric on the space of pure states of the noncommutative algebra generated by the coordinates. This metric generalizes the usual Riemannian one. We investigate some general properties of this metric in the finite ..."
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Cited by 18 (8 self)
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Following the general principles of noncommutative geometry, it is possible to define a metric on the space of pure states of the noncommutative algebra generated by the coordinates. This metric generalizes the usual Riemannian one. We investigate some general properties of this metric in the finite commutative case which corresponds to a metric on a finite set and also give some examples of computations in both commutative and noncommutative cases. PACS99: 04.60.Nc Lattice and discrete methods. Number of figures: 2
Orderunit quantum GromovHausdorff distance
 J. Funct. Anal
, 2003
"... 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 sh ..."
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Cited by 13 (5 self)
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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 θdeformations of Connes and Landi are continuous in the parameter θ. 1.
Commutative geometries are spin manifolds
 Rev. Math. Phys
"... In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spin c geometry depending on whether the geometry is “real ” or not. We attempt to flesh out the details of Connes ’ ideas. As an illustr ..."
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Cited by 13 (1 self)
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In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spin c geometry depending on whether the geometry is “real ” or not. We attempt to flesh out the details of Connes ’ ideas. As an illustration we present a proof of his claim, partly extending the validity of the result to pseudoRiemannian spin manifolds. Throughout we are as explicit and elementary as possible. 1
Hyperbolic group C ∗ algebras and freeproduct C ∗ algebras as compact quantum metric spaces
 Canad. J. Math
"... 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). W ..."
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Cited by 12 (2 self)
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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 wordlength 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 “ Haageruptype ” 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 muchstudied class of “compact noncommutative spaces ” (that is, unital C ∗algebras). In [3] Connes showed that the “Dirac ” operator of a spectral triple (i.e.