# Quantum expanders and geometry of operator spaces

### Gilles Pisier

Texas A&M University, College Station, USA

## 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 $expβnN_{2}$ 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.

## Cite this article

Gilles Pisier, Quantum expanders and geometry of operator spaces. J. Eur. Math. Soc. 16 (2014), no. 6, pp. 1183–1219

DOI 10.4171/JEMS/458