Related papers: Bounded-Lipschitz Distances on the State Space of …
By a quantum metric space we mean a C^*-algebra (or more generally an order-unit space) equipped with a generalization of the Lipschitz seminorm on functions which is defined by an ordinary metric. We develop for compact quantum metric…
A compact quantum metric space is a unital $C^*$-algebra equipped with a Lip-norm. Let $\{(A_n, L_n)\}$ be a sequence of compact quantum metric spaces, and let $\phi_n:A_n\to A_{n+1}$ be a unital $^*$-homomorphism preserving Lipschitz…
We prove that the topology on the density space with respect to a unital C*-algebra and a faithful induced by the C*-norm is finer than the Bures metric topology. We also provide an example when this containment is strict. Next, we provide…
On a discrete group G a length function may implement a spectral triple on the reduced group C*-algebra. Following A. Connes, the Dirac operator of the triple then can induce a metric on the state space of reduced group C*-algebra. Recent…
It is proved that the family of equivalence classes of Lip-normed C*-algebras introduced by M. Rieffel, up to isomorphisms preserving the Lip-seminorm, is not complete w.r.t. the matricial quantum Gromov-Hausdorff distance introduced by D.…
We introduce a new distance dist_oq between compact quantum metric spaces. We show that dist_oq is Lipschitz equivalent to Rieffel's distance dist_q, and give criteria for when a parameterized family of compact quantum metric spaces is…
For a closed cocompact subgroup $\Gamma$ of a locally compact group $G$, given a compact abelian subgroup $K$ of $G$ and a homomorphism $\rho:\hat{K}\to G$ satisfying certain conditions, Landstad and Raeburn constructed equivariant…
We introduce a new metric on the ideal space of an AF algebra that metrizes the Fell topology. The novelty of this metric lies in the use of a Hamming distance type metric in its construction. Furthermore, this metric captures more of the…
Given a compact metric space X and a unital AF algebra A equipped with a faithful tracial state, we place quantum metrics on the tensor product of C(X) and A given established quantum metrics on C(X) and A from work with Bice and…
We use compactifications of C*-algebras to introduce noncommutative coarse geometry. We transfer a noncommutative coarse structure on a C*-algebra with an action of a locally compact Abelian group by translations to Rieffel deformations and…
In this work, a metric is presented on the set of boundedly-compact pointed metric spaces that generates the Gromov-Hausdorff topology. A similar metric is defined for measured metric spaces that generates the Gromov-Hausdorff-Prokhorov…
Marc Rieffel had introduced the notion of the quantum Gromov-Hausdorff distance on compact quantum metric spaces and found a sequence of matrix algebras that converges to the space of continuous functions on $2$-sphere in this distance. One…
A distance between von Neumann algebras is introduced, depending on a further norm inducing the $w^*$-topology on bounded sets. Such notion is related both with the Gromov-Hausdorff distance for quantum metric spaces of Rieffel and with the…
We use the theory of quantization to introduce non-commutative versions of metric on state space and Lipschitz seminorm. We show that a lower semicontinuous matrix Lipschitz seminorm is determined by their matrix metrics on the matrix state…
We show that the noncommutative differential geometry of quantum projective spaces is compatible with Rieffel's theory of compact quantum metric spaces. This amounts to a detailed investigation of the Connes metric coming from the unital…
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…
We introduce the notion of a quantum locally compact metric space, which is the noncommutative analogue of a locally compact metric space, and generalize to the nonunital setting the notion of quantum metric spaces introduced by Rieffel. We…
We construct a new version of the dual Gromov--Hausdorff propinquity that is sensitive to the strongly Leibniz property. In particular, this new distance is complete on the class of strongly Leibniz quantum compact metric spaces. Then,…
We study some topological spaces that can be considered as hyperspaces associated to noncommutative spaces. More precisely, for a NC compact space associated to a unital C*-algebra, we consider the set of closed projections of the second…
Let $\ell$ be a length function on a group G, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following Connes, $M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a…