Geodesic
Quantum expanders and geometry of operator spaces
Journal of the European Mathematical Society, Tome 16 (2014) no. 6, pp. 1183-1219

Voir la notice de l'article provenant de la source EMS Press

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 MN​-spaces needed to represent (up to a constant C>1) the MN​-version of the n-dimensional operator Hilbert space OHn​ as a direct sum of copies of MN​. We show that, when C is close to 1, this multiplicity grows as expβnN2 for some constant β>0. 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 n-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.
DOI : 10.4171/jems/458
Classification : 46-XX, 47-XX
Keywords: Quantum expander, operator space, completely bounded map, smooth point 
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Gilles Pisier. Quantum expanders and geometry of operator spaces. Journal of the European Mathematical Society, Tome 16 (2014) no. 6, pp. 1183-1219. doi: 10.4171/jems/458

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