We give a brief survey of many of the high-lights of our present understanding of the young subject of quantum metric spaces, and of quantum Gromov-Hausdorff distance between them. We include examples. My interest in developing the theory of compact quantum metric spaces was stimulated by certain statements in the high-energy physics and string-theory literature, concerning non-commutative spac...

A large class of noncommutative spherical manifolds was obtained recently from cohomology considerations. A one-parameter family of twisted 3-spheres was discovered by Connes and Landi, and later generalized to a three-parameter family by Connes and Dubois-Violette. The spheres of Connes and Landi were shown to be homogeneous spaces for certain compact quantum groups. Here we investigate whethe...

LogicWeb has traditionally lacked devices for dealing with intractable queries. We address this limitation by adopting length-bounded inference, a form of approximate reasoning. A length-bounded inference is of the form pv(P, G, n) which is a success if a query G can be proved from the web page P within n proof steps. It thus makes LogicWeb decidable and more tractable. During the process, we p...

One of the key ingredients of A. Connes’ noncommutative geometry is a generalized Dirac operator which induces a metric(Connes’ distance) on the state space. We generalize such a Dirac operator devised by A. Dimakis et al , whose Connes’ distance recovers the linear distance on a 1D lattice, into 2D lattice. This Dirac operator being “naturally” defined has the “local eigenvalue property” and i...

Abstract. We reformulate the Baum-Connes conjecture with coe cients by introducing a new crossed product functor for C⇤-algebras. All confirming examples for the original Baum-Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to...

Let A be a Banach algebra and X a Banach A-bimodule, the derivation D : A → X is semi-inner if there are ξ, μ ∈ X such that D(a) = a.ξ − μ.a, (a ∈ A). A is called semi-amenable if every derivation D : A → X∗ is semi-inner. The dual Banach algebra A is Connes semi-amenable (resp. approximately semi-amenable) if, every D ∈ Z1w _ (A,X), for each normal, dual Banach A-bimodule X, is semi -inner (re...

We study learning in Bayesian games (or games with differential information) with an arbitrary number of bounded rational players, i.e., players who choose approximate best response strategies [approximate Bayesian Nash Equilibrium (BNE) strategies] and who also are allowed to be completely irrational in some states of the world. We show that bounded rational players by repetition can reach a l...

Bounded Max-Sum is a message-passing algorithm for solving Distributed Constraint Optimization Problems able to compute solutions with a guaranteed approximation ratio. Although its approximate solutions were empirically proved to be within a small percentage of the optimal solution on low and moderately dense problems, in this paper we show that its theoretical approximation ratio is overestim...