# Quelques propriétés des algèbres de von Neumannengendrées par des q-Gaussiens

Abstract : This work is at the crossroads of operator algebra and
non-commutative probability theories. Some properties of
$\Gamma_{q}(H_{\R})$ are investigated, where $\Gamma_{q}(H_{\R})$
stands for the von Neumann algebra generated by non-commutative
$q$-deformed Gaussian variables . These variables are given as
operators acting on a $q-$deformed Fock space where the
$q$-canonical commutation relations are realized by non-commutative
shift operators.

Some $L^{\infty}$-Khintchin type inequalities with operator
coefficients, concerning Wick products of a given length, are
discussed and established in the first chapter. These inequalities
extend, on the one hand Haagerup's scalar inequalities in the free
case and, on the other hand Bo\.zejko and Speicher's operator
coefficients inequalities for $q$-Gaussians. From those inequalities
follows the non-injectivity of $\Gamma_{q}(H_{\R})$ as soon as
$\dim_{\R}(H_{\R})> 1$.

The second chapter is devoted to the construction of an asymptotic
matricial model for $q$-Gaussian variables. Such a model is then
used to prove that all $q$-Gaussian algebras are QWEP.

The $C^*-$algebraic case is also investigated and, the preceding
results are studied and stated for various other generalized
$q$-Gaussian algebras such as type $I\!I\!I$ $q$-Gaussian algebras
and $T-$deformed Gaussian algebras where $T$ is a Yang-Baxter
operator.
Keywords :
Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00077616
Contributor : Alexandre Nou <>
Submitted on : Wednesday, May 31, 2006 - 2:33:00 PM
Last modification on : Thursday, January 28, 2021 - 10:26:02 AM
Long-term archiving on: : Tuesday, September 18, 2012 - 2:30:10 PM

### Identifiers

• HAL Id : tel-00077616, version 1

### Citation

Alexandre Nou. Quelques propriétés des algèbres de von Neumann
engendrées par des q-Gaussiens. Mathématiques [math]. Université de Franche-Comté, 2004. Français. ⟨tel-00077616⟩

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