Quantum expanders and geometry of operator spaces

Source 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 OH_n as a direct sum of copies of M_N . We show that, when C is close to 1, this multiplicity grows as \exp{\beta n N^2} for some constant \beta>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.