## Factoriality of q-gaussian von neumann algebras

Venue: | Comm. Math. Phys |

Citations: | 8 - 1 self |

### BibTeX

@ARTICLE{Ricard_factorialityof,

author = {Éric Ricard},

title = {Factoriality of q-gaussian von neumann algebras},

journal = {Comm. Math. Phys},

year = {},

pages = {659--665}

}

### OpenURL

### Abstract

We prove that the von Neumann algebras generated by n q-Gaussian elements, are factors for n � 2. 1

### Citations

65 |
An example of a generalized Brownian motion
- Bo˙zejko, Speicher
- 1991
(Show Context)
Citation Context ...ators satisfying the q-canonical commutation relations, for −1 < q < 1 : l(e)l ∗ (f) − ql ∗ (f)l(e) = (e, f)Id. Nevertheless their existence was proved only 20 years later by Bo˙zejko and Speicher in =-=[2]-=-. Since, many people studied the von Neumann algebra Γq(HR), generated by qGaussian random variables {l(e)+l ∗ (e); e ∈ HR}, and some of their generalizations. It is well know that Γq(HR) is type II1.... |

64 | q-Gaussian processes: noncommutative and classical aspects
- Bo˙zejko, Kümmerer, et al.
- 1997
(Show Context)
Citation Context ... H, which was removed by Nou [4]. Recently, Shlyakhtenko [5] proved that they are solid for some values of q. The question of the factoriality of Γq(HR) was studied by Bo˙zejko, Kümmerer and Speicher =-=[1]-=-, they showed that if H is infinite dimensional then Γq(HR) is a factor. This condition was partially released by ´ Sniady [6], who showed that this is still true if the dimension of H is greater than... |

30 |
Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces
- Bo˙zejko, Speicher
- 1994
(Show Context)
Citation Context ... that this is still true if the dimension of H is greater than a function of q. 2 Preliminaries In this paper, −1 < q < 1 is a fixed real number, we will use standard notation and refer to the papers =-=[3, 1, 4]-=- for general background. Let H be the complexification of some real Hilbert space HR. By H⊗2n (n � 1), we denote the hilbertian n-tensor product of H with itself, this space is equipped with a scalar ... |

13 | Some estimates for non–microstates free entropy dimension with applications to q–semicircular families
- Shlyakhtenko
(Show Context)
Citation Context ...probability theory. Among the known results, Bo˙zejko and Speicher showed that Γq(HR) is non injective under some condition on the dimension of H, which was removed by Nou [4]. Recently, Shlyakhtenko =-=[5]-=- proved that they are solid for some values of q. The question of the factoriality of Γq(HR) was studied by Bo˙zejko, Kümmerer and Speicher [1], they showed that if H is infinite dimensional then Γq(H... |

8 |
Non injectivity of the q-deformed von Neumann algebra
- Nou
(Show Context)
Citation Context ... central object in the free probability theory. Among the known results, Bo˙zejko and Speicher showed that Γq(HR) is non injective under some condition on the dimension of H, which was removed by Nou =-=[4]-=-. Recently, Shlyakhtenko [5] proved that they are solid for some values of q. The question of the factoriality of Γq(HR) was studied by Bo˙zejko, Kümmerer and Speicher [1], they showed that if H is in... |

1 |
Factoriality of Bo˙zejko–Speicher von Neumann algebras. Math. arXiv OA/0307201. Éric Ricard Département de Mathématiques de Besançon Université de Franche-Comté 25030 Besançon cedex e-mail: eric.ricard@math.univ-fcomte.fr
- Piotr
(Show Context)
Citation Context ...the factoriality of Γq(HR) was studied by Bo˙zejko, Kümmerer and Speicher [1], they showed that if H is infinite dimensional then Γq(HR) is a factor. This condition was partially released by ´ Sniady =-=[6]-=-, who showed that this is still true if the dimension of H is greater than a function of q. 2 Preliminaries In this paper, −1 < q < 1 is a fixed real number, we will use standard notation and refer to... |