2,241 research outputs found
A Compactness Theorem for The Dual Gromov-Hausdorff Propinquity
We prove a compactness theorem for the dual Gromov-Hausdorff propinquity as a
noncommutative analogue of the Gromov compactness theorem for the
Gromov-Hausdorff distance. Our theorem is valid for subclasses of quasi-Leibniz
compact quantum metric spaces of the closure of finite dimensional
quasi-Leibniz compact quantum metric spaces for the dual propinquity. While
finding characterizations of this class proves delicate, we show that all
nuclear, quasi-diagonal quasi-Leibniz compact quantum metric spaces are limits
of finite dimensional quasi-Leibniz compact quantum metric spaces. This result
involves a mild extension of the definition of the dual propinquity to
quasi-Leibniz compact quantum metric spaces, which is presented in the first
part of this paper.Comment: 40 Pages. Version 4 includes several minor corrections and is
accepted in the Indiana University Mathematics Journa
Convergence of Fuzzy Tori and Quantum Tori for the quantum Gromov-Hausdorff Propinquity: an explicit approach
Quantum tori are limits of finite dimensional C*-algebras for the quantum
Gromov-Hausdorff propinquity, a metric defined by the author as a strengthening
of Rieffel's quantum Gromov-Hausdorff designed to retain the C*-algebraic
structure. In this paper, we propose a proof of the continuity of the family of
quantum and fuzzy tori which relies on explicit representations of the
C*-algebras rather than on more abstract arguments, in a manner which takes
full advantage of the notion of bridge defining the quantum propinquity.Comment: 41 Pages. This paper is the second half of ArXiv:1302.4058v2. The
latter paper has been divided in two halves for publications purposes, with
the first half now the current version of 1302.4058, which has been accepted
in Trans. Amer. Math. Soc. This second half is now a stand-alone paper, with
a brief summary of 1302.4058 and a new introductio
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
Differentiable structures on metric measure spaces: A Primer
This is an exposition of the theory of differentiable structures on metric
measures spaces, in the sense of Cheeger and Keith.Comment: 23 page
The Quantum Gromov-Hausdorff Propinquity
We introduce the quantum Gromov-Hausdorff propinquity, a new distance between
quantum compact metric spaces, which extends the Gromov-Hausdorff distance to
noncommutative geometry and strengthens Rieffel's quantum Gromov-Hausdorff
distance and Rieffel's proximity by making *-isomorphism a necessary condition
for distance zero, while being well adapted to Leibniz seminorms. This work
offers a natural solution to the long-standing problem of finding a framework
for the development of a theory of Leibniz Lip-norms over C*-algebras.Comment: 49 Pages. This is the first half of 1302.4058v2, which has been
accepted in Trans. Amer. Math. Soc. The second half is now a different paper
entitled "Convergence of Fuzzy Tori and Quantum Tori for the quantum
Gromov-Hausdorff Propinquity: an explicit approach
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