Quantum expanders and geometry of operator spaces

Abstract

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 M N-spaces needed to represent (up to a constant C > 1) the M N-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 exp(βnN2) 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. © European Mathematical Society 2014.