Matricial quantum Gromov–Hausdorff distance
By David Kerr. 2002
84 citations recorded by the source corpus.
Abstract We develop a matricial version of Rieffel's Gromov–Hausdorff distance for compact quantum metric spaces within the setting of operator systems and unital C ∗ -algebras. Our approach yields a metric space of “isometric” unital complete order isomorphism classes of metrized operator systems which in many cases exhibits the same convergence properties as those in the quantum metric setting, as for example in Rieffel's approximation of the sphere by matrix algebras using Berezin quantization. Within the metric subspace of metrized unital C ∗ -algebras we establish the convergence of sequences which are Cauchy with respect to a larger Leibniz distance, and we also prove an analogue of the precompactness theorems of Gromov and Rieffel.
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/a5314d819b27ba0136dcc4e83333f04a4315ab7a
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In Algebra and Number Theory
In Mathematical Physics
Citations in the corpus
- Quantum Ultrametrics on AF Algebras and The Gromov-Hausdorff Propinquity (2015, unknown): “The quasi-Leibniz property is the second author’s added requirement, itself largely based on Kerr’s similar notion of the F -Leibniz property [15], but used for a very different reason — the quasi-Leibniz property is used to ensure that the Gromov-Hausdorff propinquity has the desired coincidence pr”
- The Quantum Gromov-Hausdorff Propinquity (2013, background): “t for Rieffel’s proximity, it is unclear whether the sequence is Cauchy, and this leads to intriguing questions [42]. This is not the first time such problems occur: Kerr already faced this problem in [25]. The main problem can be summarized in the fact that the quotient of a Leibniz seminorm is not ”
- Gromov-Hausdorff Distance for Quantum Metric Spaces (2000, background): “Very recently, building on the present paper, Hanfeng Li has shown [46] for the Connes-Landi-DuboisViolette spheres {Sθ} (and related quantum manifolds) [17] [ 16 ] with their Dirac operators, that they form compact quantum metric spaces, and that if a sequence θn of parameters converges to a parame”