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

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DOI

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 
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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     title = {Quantum expanders and geometry of operator spaces},
     journal = {Journal of the European Mathematical Society},
     pages = {1183--1219},
     year = {2014},
     volume = {16},
     number = {6},
     doi = {10.4171/jems/458},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/458/}
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