The von Neumann algebras generated by t-gaussians
Annales de l'Institut Fourier, Volume 56 (2006) no. 2, pp. 475-498.

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.

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.

DOI: 10.5802/aif.2190
Classification: 46L54,  46L10
Keywords: Conditionnal free product, interacting Fock space.
Ricard, Éric 1

1 Laboratoire de Mathématiques Université de Franche-Comté 25030 Besançon, cedex (France)
@article{AIF_2006__56_2_475_0,
     author = {Ricard, \'Eric},
     title = {The von {Neumann} algebras generated by $t$-gaussians},
     journal = {Annales de l'Institut Fourier},
     pages = {475--498},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {2},
     year = {2006},
     doi = {10.5802/aif.2190},
     mrnumber = {2226024},
     zbl = {1116.46056},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2190/}
}
TY  - JOUR
AU  - Ricard, Éric
TI  - The von Neumann algebras generated by $t$-gaussians
JO  - Annales de l'Institut Fourier
PY  - 2006
DA  - 2006///
SP  - 475
EP  - 498
VL  - 56
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2190/
UR  - https://www.ams.org/mathscinet-getitem?mr=2226024
UR  - https://zbmath.org/?q=an%3A1116.46056
UR  - https://doi.org/10.5802/aif.2190
DO  - 10.5802/aif.2190
LA  - en
ID  - AIF_2006__56_2_475_0
ER  - 
%0 Journal Article
%A Ricard, Éric
%T The von Neumann algebras generated by $t$-gaussians
%J Annales de l'Institut Fourier
%D 2006
%P 475-498
%V 56
%N 2
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2190
%R 10.5802/aif.2190
%G en
%F AIF_2006__56_2_475_0
Ricard, Éric. The von Neumann algebras generated by $t$-gaussians. Annales de l'Institut Fourier, Volume 56 (2006) no. 2, pp. 475-498. doi : 10.5802/aif.2190. http://archive.numdam.org/articles/10.5802/aif.2190/

[1] Accardi, L.; Bożejko, M. Interacting Fock spaces and Gaussianization of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Volume 4 (1998) no. 1, pp. 663-670 | DOI | MR | Zbl

[2] Boca, F. Free products of completely positive maps and spectral sets, J. Funct. Anal., Volume 97 (1991) no. 2, pp. 251-263 | DOI | MR | Zbl

[3] Boca, F. Completely positive maps on amalgamated product C * -algebras, Math. Scand., Volume 72 (1993) no. 2, pp. 212-222 | MR | Zbl

[4] Bożejko, M.; Fendler, G. A note on certain partial sum operators (Preprint) | Zbl

[5] Bożejko, M.; Kümmerer, B.; Speicher, R. q-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys., Volume 185 (1997) no. 1, pp. 129-154 | DOI | MR | Zbl

[6] Bożejko, M.; Leinert, M.; Speicher, R. Convolution and limit theorems for conditionally free random variables, Pacific J. Math., Volume 175 (1996) no. 2, pp. 357-388 | MR | Zbl

[7] Bożejko, M.; Speicher, R. Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces, Math. Ann., Volume 300 (1994) no. 1, pp. 97-120 | DOI | MR | Zbl

[8] Bożejko, M.; Wysoczański, J. Remarks on t-transformations of measures and convolutions, Ann. Inst. H. Poincaré Probab. Statist., Volume 37 (2001) no. 6, pp. 737-761 | DOI | Numdam | MR | Zbl

[9] Buchholz, A. L -Khintchine-Bonami inequality in free probability, Quantum probability (Gdańsk, 1997) (Banach Center Publ.), Volume 43, Polish Acad. Sci., Warsaw, 1998, pp. 105-109 | MR | Zbl

[10] Dykema, K. J. Faithfulness of free product states, J. Funct. Anal., Volume 154 (1998) no. 2, pp. 323-329 | DOI | MR | Zbl

[11] Haagerup, U. An example of a nonnuclear C * -algebra, which has the metric approximation property, Invent. Math., Volume 50 (1978/79) no. 3, pp. 279-293 | DOI | MR | Zbl

[12] Młotkowski, W. Operator-valued version of conditionally free product, Studia Math., Volume 153 (2002) no. 1, pp. 13-30 | DOI | MR | Zbl

[13] Nou, A. Non injectivity of the q-deformed von Neumann algebra, Math. Ann., Volume 330 (2004) no. 1, pp. 17-38 | DOI | MR | Zbl

[14] Ricard, É. Factoriality of q-Gaussian von Neumann Algebras, Comm. Math. Phys., Volume 257 (2005) no. 2, pp. 659-665 | DOI | MR | Zbl

[15] Shlyakhtenko, D. Some estimates for non-microstates free entropy dimension with applications to q-semicircular families, Int. Math. Res. Not., Volume 51 (2004), pp. 2757-2772 | DOI | MR | Zbl

[16] Voiculescu, D. V.; Dykema, K. J.; Nica, A. Free random variables, CRM Monograph Series, 1, American Mathematical Society, Providence, RI, 1992 (A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups) | MR | Zbl

[17] Wojakowski, Ł. Probabilistyka interpolujaca pomiedzy wolna boolowska, Wrocław University (2004) (Ph. D. Thesis)

[18] Wysoczański, J. The von Neumann algebra associated with t-free non-commutative gaussian random variables (notes)

Cited by Sources: