2004/11/08 by Kerr, David, Li, Hanfeng · 1 citation
#46L07 #46L87 #53C23 #58B34 #FOS: Mathematics #Metric Geometry (math.MG) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.math/0411157
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.