Quantum expanders and geometry of operator spaces

Pisier, Gilles. "Quantum expanders and geometry of operator spaces." Journal of the European Mathematical Society 016.6 (2014): 1183-1219. <http://eudml.org/doc/277784>.

@article{Pisier2014,
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.},
author = {Pisier, Gilles},
journal = {Journal of the European Mathematical Society},
keywords = {quantum expander; operator space; completely bounded map; smooth point; operational Hilbert space; completely bounded map; operator space; quantum expander; smooth point; operational Hilbert space},
language = {eng},
number = {6},
pages = {1183-1219},
publisher = {European Mathematical Society Publishing House},
title = {Quantum expanders and geometry of operator spaces},
url = {http://eudml.org/doc/277784},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Pisier, Gilles
TI - Quantum expanders and geometry of operator spaces
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 6
SP - 1183
EP - 1219
AB - 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.
LA - eng
KW - quantum expander; operator space; completely bounded map; smooth point; operational Hilbert space; completely bounded map; operator space; quantum expander; smooth point; operational Hilbert space
UR - http://eudml.org/doc/277784
ER -