We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. We first show that on Riemannian spin manifolds the two distances coincide. Then, on convex manifolds in the sense of Nash… (More)
We construct a family of self-adjoint operators DN , N ∈ Z, which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space CP q , for any ≥ 2 and 0 < q < 1. They provide 0 +-dimensional equivariant even spectral triples. If is odd and N = 1 2 (+ 1), the spectral triple is real with KO-dimension 2 mod 8.
We construct spectral triples on all Podle´s quantum spheres S 2 qt. These noncom-mutative geometries are equivariant for a left action of U q (su(2)) and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the sphere S 2. There is also an equivariant real structure for which both the commutant property… (More)
We discuss spectral properties of the equatorial Podle´s sphere S 2 q. As a preparation we also study the 'degenerate' (i.e. q = 0) case (related to the quantum disk). Over S 2 q we consider two different spectral triples: one related to the Fock representation of the Toeplitz algebra and the isopectral one given in . After the identification of the… (More)
Equivariance under the action of U q (so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the quantum Euclidean 4-sphere S 4 q. These representations are the constituents of a spectral triple on S 4 q with a Dirac operator which is isospectral to the canonical one of the spin structure of the round sphere S 4… (More)
We construct explicit generators of the K-theory and K-homology for the coordinate algebra of 'functions' on the quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Dirac-like operators and spectral triples of any positive real dimension.
Equivariance under the action of U q (so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the orthogonal quantum 4-sphere S 4 q. These representations are the constituents of a spectral triple on S 4 q with a Dirac operator which is isospectral to the canonical one on the round sphere S 4 and which then gives… (More)
Through the example of the quantum symplectic 4-sphere, we discuss how the notion of twisted spectral triple fits into the framework of quantum homogeneous spaces.
In this thesis I study various aspects of theories in the two most studied examples of non-commutative spacetimes: canonical spacetime ([x µ , x ν ] = θ µν) and κ-Minkowski spacetime ([x i , t] = κ −1 x i). In the first part of the thesis I consider the description of the propagation of " classical " waves in these spacetimes. In the case of κ-Minkowski… (More)
We describe how to obtain the imprimitivity bimodules of the noncommutative torus from a " principal bundle " construction, where the total space is a quasi-associative deformation of a 3-dimensional Heisenberg manifold.