Corpus ID: 119161930

# The Gromov-Hausdorff propinquity for metric Spectral Triples.

```@article{Latrmolire2018TheGP,
title={The Gromov-Hausdorff propinquity for metric Spectral Triples.},
author={Fr{\'e}d{\'e}ric Latr{\'e}moli{\`e}re},
journal={arXiv: Operator Algebras},
year={2018}
}```
• F. Latrémolière
• Published 27 November 2018
• Mathematics, Physics
• arXiv: Operator Algebras
We define a metric on the class of metric spectral triples, which is null exactly between spectral triples with unitary equivalent Dirac operators and *-isomorphic underlying C*-algebras. This metric dominates the propinquity, and thus implies metric convergence of the {\qcms s} induced by metric spectral triples. In the process of our construction, we also introduce the covariant modular propinquity, as a key component for the definition of the spectral propinquity.
3 Citations

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#### References

SHOWING 1-10 OF 33 REFERENCES
Noncommutative Solenoids and the Gromov-Hausdorff Propinquity
• Mathematics
• 2016
We prove that noncommutative solenoids are limits, in the sense of the Gromov-Hausdorff propinquity, of quantum tori. From this observation, we prove that noncommutative solenoids can be approximatedExpand
Dirac operators and spectral triples for some fractal sets built on curves
• Mathematics
• 2006
Abstract We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples whereExpand
CONNES DISTANCE BY EXAMPLES: HOMOTHETIC SPECTRAL METRIC SPACES
We study metric properties stemming from the Connes spectral distance on three types of non-compact non-commutative spaces which have received attention recently from various viewpoints in theExpand
Quantum Ultrametrics on AF Algebras and The Gromov-Hausdorff Propinquity
• Mathematics
• 2015
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 finiteExpand
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-HausdorffExpand
Matrix geometries and fuzzy spaces as finite spectral triples
A class of real spectral triples that are similar in structure to a Riemannian manifold but have a finite-dimensional Hilbert space is defined and investigated, determining a general form for theExpand
Gromov-Hausdorff Distance for Quantum Metric Spaces
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.Expand
Spectral geometry with a cut-off: topological and metric aspects
• Mathematics, Physics
• 2014
Abstract Inspired by regularization in quantum field theory, we study topological and metric properties of spaces in which a cut-off is introduced. We work in the framework of noncommutativeExpand
Approximation of quantum tori by finite quantum tori for the quantum Gromov–Hausdorff distance☆
We establish that, given a compact Abelian group G endowed with a continuous length function l and a sequence (Hn)n∈N of closed subgroups of G converging to G for the Hausdorff distance induced by l,Expand
Leibniz seminorms for "Matrix algebras converge to the sphere"
In an earlier paper of mine relating vector bundles and Gromov-Hausdorff distance for ordinary compact metric spaces, it was crucial that the Lipschitz seminorms from the metrics satisfy a strongExpand