We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the ``growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of -spaces needed to represent (up to a constant ) the -version of the -dimensional operator Hilbert space as a direct sum of copies of . We show that, when is close to 1, this multiplicity grows as for some constant . The main idea is to relate quantum expanders with "smooth" points on the matricial analogue of the Euclidean unit sphere. This generalizes to operator spaces a classical geometric result on -dimensional Hilbert space (corresponding to N=1). In an appendix, we give a quick proof of an inequality (related to Hastings's previous work) on random unitary matrices that is crucial for this paper.
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Gilles Pisier, Quantum expanders and geometry of operator spaces. J. Eur. Math. Soc. 16 (2014), no. 6, pp. 1183–1219