1209.2059 (Gilles Pisier)

Quantum Expanders and Geometry of Operator Spaces    [PDF]

Gilles Pisier

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 "local theory" of operator spaces. This allows us to provide 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 identify quantum expanders with "smooth" points on the matricial analogue of the unit sphere. This generalizes to operator spaces a classical geometric result on $n$-dimensional Hilbert space (corresponding to N=1). Our work strongly suggests to further study a certain class of operator spaces that we call matricially subGaussian. In a second part, we introduce and study a generalization of the notion of exact operator space that we call subexponential. Using Random Matrices we show that the factorization results of Grothendieck type that are known in the exact case all extend to the subexponential case, and we exhibit (a continuum of distinct) examples of non-exact subexponential operator spaces. We also show that $OH$, $R+C$ and $\max(\ell_2)$ (or any other maximal operator space) are not subexponential.

View original: http://arxiv.org/abs/1209.2059