On Gromov-Hausdorff convergence for operator metric spaces

By David Kerr, Hanfeng Li. 2004

61 citations recorded by the source corpus.

We introduce an analogue for Lip-normed operator systems of the second author's order-unit quantum Gromov-Hausdorff distance and prove that it is equal to the first author's complete distance. This enables us to consolidate the basic theory of what might be called operator Gromov-Hausdorff convergence. In particular we establish a completeness theorem and deduce continuity in quantum tori, Berezin-Toeplitz quantizations, and theta-deformations from work of the second author. We show that approximability by Lip-normed matrix algebras is equivalent to 1-exactness of the underlying operator space and, by applying a result of Junge and Pisier, that for n greater than or equal to 7 the set of isometry classes of n-dimensional Lip-normed operator systems is nonseparable. We also treat the question of generic complete order structure.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/34862f73f00bdf76f9a782e8f8b3d9e28be7efe5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Nuclearity Related Properties in Operator Systems (2011, methodology): “The amalgamated sum of two operator systems over the unit introduced in [23] (or coproduct of two operator systems in the language of [13]) seem to be another natural structure to seek the stability of several nuclearity properties.”
Characterisations of the weak expectation property (2013, background): “Li [18], where the authors described the amalgamation process over a joint operator subsystem.”
The Quantum Gromov-Hausdorff Propinquity (2013, methodology): “A natural modification of Li’s nuclear distance, the unital nuclear distance introduced in [26], does dominate the quantum propinquity (and Li’s original nuclear distance).”