The von Neumann algebras generated by $t$-gaussians

Ricard, Éric

Annales de l'Institut Fourier, Tome 56 (2006) no. 2, pp. 475-498.

Dans la théorie des probabilités non commutative, beaucoup de déformations ou généralisations de la notion de produit libre sont apparues, comme les concepts de probabilités libres conditionnelles et d’espaces de Fock interactifs. L’un des premiers exemples d’algèbres ainsi obtenu est l’objet de cet article  : les algèbres de von Neumann engendrées par un nombre fini $n$ d’opérateurs $t$-gaussiens. Il s’avère qu’à $n$ fixé, si $t$ est suffisamment proche de $1$, alors ces algèbres ne dépendent pas de $t$. Plus généralement, on donne une condition qui assure un isomorphisme entre un produit libre conditionnel et un produit libre réduit usuel.

We study the $t$-deformation of gaussian von Neumann algebras. They appear as example in the theories of Interacting Fock spaces and conditionally free products. When the number of generators is fixed, it is proved that if $t$ is sufficiently close to $1$, then these algebras do not depend on $t$. In the same way, the notion of conditionally free von Neumann algebras often coincides with freeness.

Classification : 46L54, 46L10
Mots clés : Conditionnal free product, interacting Fock space.

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