# Quantum expanders and geometry of operator spaces

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 6, page 1183-1219
- ISSN: 1435-9855

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topPisier, 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 -

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