27 research outputs found
The Triangle Inequality and the Dual Gromov-Hausdorff Propinquity
he dual Gromov-Hausdorff propinquity is a generalization of the Gromov-Hausdorff distance to the class of Leibniz quantum compact metric spaces, designed to be well-behaved with respect to C*-algebraic structures. In this paper, we present a variant of the dual propinquity for which the triangle inequality is established without the recourse to the notion of journeys, or finite paths of tunnels. Since the triangle inequality has been a challenge to establish within the setting of Leibniz quantum compact metric spaces for quite some time, and since journeys can be a complicated tool, this new form of the dual propinquity is a significant theoretical and practical improvement. On the other hand, our new metric is equivalent to the dual propinquity, and thus inherits all its properties
Curved Noncommutative Tori as Leibniz Quantum Compact Metric Spaces
We prove that curved noncommutative tori, introduced by Dabrowski and Sitarz,
are Leibniz quantum compact metric spaces and that they form a continuous
family over the group of invertible matrices with entries in the commutant of
the quantum tori in the regular representation, when this group is endowed with
a natural length function.Comment: 16 Pages, v3: accepted in Journal of Math. Physic
Symmetry in the Cuntz Algebra on two generators
We investigate the structure of the automorphism of which
exchanges the two canonical isometries. Our main observation is that the fixed
point C*-subalgebra for this action is isomorphic to and we
detail the relationship between the crossed-product and fixed point subalgebra.Comment: 14 Pages. Minor changes and additions to the references sectio
Characterization of the Sequential Product on Quantum Effects
We present a characterization of the standard sequential product of quantum
effects. The characterization is in term of algebraic, continuity and duality
conditions that can be physically motivated.Comment: 11 pages. Accepted for publication in the Journal of Mathematical
Physic
Quantum Locally Compact Metric Spaces
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 then provide several examples of such structures, including the
Moyal plane, as well as compact quantum metric spaces and locally compact
metric spaces. This paper provides an answer to the question raised in the
literature about the proper notion of a quantum metric space in the nonunital
setup and offers important insights into noncommutative geometry for non
compact quantum spaces.Comment: 39 Pages. Changes from v1: Many minor typos corrected, improved
Theorem 3.1
A Survey of the Dual Gromov-Hausdorff Propinquity
We present a survey of the dual Gromov-Hausdorff propinquity, a noncommutative analogue of the Gromov-Hausdorff distance which we introduced to provide a framework for the study of the noncommutative metric properties of C*-algebras. We first review the notions of quantum locally compact metric spaces, and present various examples of such structures. We then explain the construction of the dual Gromov-Hausdorff propinquity, first in the context of quasiLeibniz quantum compact metric spaces, and then in the context of pointed proper quantum metric spaces. We include a few new result concerning perturbations of the metrics on Leibniz quantum compact metric spaces in relation with the dual Gromov-Hausdorff propinquity