Journal of Operator Theory

Volume 62, Issue 1, Summer 2009 pp. 83-109.

On Gromov-Hausdorff convergence for operator metric spaces

Authors: David Kerr (1) and Hanfeng Li (2)
Author institution: (1) Department of Mathematics, Texas A&M University, College Station TX 77843-3368, U.S.A.
(2) Department of Mathematics, SUNY at Buffalo, Buffalo NY 14260-2900, U.S.A.

Summary: We introduce an analogue for Lip-normed operator systems of the second author's order-unit quantum Gromov--Hausdorff distance and\break 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 $n\geqslant 7$ the set of isometry classes of $n$-dimensional Lip-normed operator systems is nonseparable. We also treat the question of generic complete order structure.

Keywords: Operator system, Lip-norm, Gromov--Hausdorff distance.

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