On Gromov-Hausdorff convergence for operator metric spaces

Details On Gromov-Hausdorff convergence for operator metric spaces On Gromov-Hausdorff convergence for operator metric spaces

2004-11-08

arxiv.org/abs/math/0411157v4

Mathematics

Operator Algebras

23 pages; Section 7 added and Section 2 expanded; to appear in J. Operator Theory

Scientific paper

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.

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