Factoriality of q-Gaussian von Neumann Algebras

Ricard, Éric

doi:10.1007/s00220-004-1266-5

Cited by 54 publications

(52 citation statements)

References 5 publications

(10 reference statements)

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“…In the α = 0 case the factoriality was proved by Bożejko, Kümmerer and Speicher [14] and by Śniady [47] in special cases and then by Ricard [42] in full generality. Moreover Nou showed that vN(G 0,q (H R )) is not injective if dim(H R ) ≥ 2 [40].…”

Section: Traciality Of the Vacuum Statementioning

confidence: 92%

Fock space associated to Coxeter groups of type B

Bożejko

Ejsmont

2015

Journal of Functional Analysis

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In this article we construct a generalized Gaussian process coming from Coxeter groups of type B. It is given by creation and annihilation operators on an (α, q)-Fock space, which satisfy the commutation relationwhere x, y are elements of a complex Hilbert space with a self-adjoint involution x →x and N is the number operator with respect to the grading on the (α, q)-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators b α,q (x) + b * α,q (x) with respect to the vacuum expectation becomes a generalized Gaussian distribution, in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions. Our generalized Gaussian distribution is associated to the orthogonal polynomials called the q-Meixner-Pollaczek polynomials, yielding the q-Hermite polynomials when α = 0 and free Meixner polynomials when q = 0.

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Section: Traciality Of the Vacuum Statementioning

confidence: 92%

Fock space associated to Coxeter groups of type B

Bożejko

Ejsmont

2015

Journal of Functional Analysis

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“…Recent works have been focused on the study of mathematical properties of q-Gaussian functions [70][71][72][73][74][75][76][77][78], including methods for generating random numbers which follow q-Gaussian distributions [79,80]. q-Gaussians have been employed in the study of a wide range of themes including probabilistic models [81,82], stellar plasmas [83], porous-medium equation [84], Bose-condensed gases [85][86][87], dynamical systems [88][89][90], polymeric networks [91], smallworld networks [92], fingering processes [93], processes with stochastic volatility [94,95] and nonlinear diffusion [96,97].…”

Section: Q-gaussian Distributionmentioning

confidence: 99%

q-distributions in complex systems: a brief review

Picoli¹,

Mendes²,

Malacarne³

et al. 2009

Braz. J. Phys.

103

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The nonextensive statistical mechanics proposed by Tsallis is today an intense and growing research field. Probability distributions which emerges from the nonextensive formalism (q-distributions) have been applied to an impressive variety of problems. In particular, the role of q-distributions in the interdisciplinary field of complex systems has been expanding. Here, we make a brief review of q-exponential, q-Gaussian and q-Weibull distributions focusing some of their basic properties and recent applications. The richness of systems analyzed may indicate future directions in this field.

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“…known as the Touchard-Riordan formula [58,56,57,49]. No analogue of the above expression is presently known for the joint generating function of crossings and nestings.…”

Section: Next Let [N] Qt Denote the (Q T)-analogue Of A Positive Imentioning

confidence: 96%

“…For any Hilbert space H of dimension n 2, Γ q (H ) is a factor[13,53,48] and Γ 0 (H ) is isomorphic to the free group factor on n generators[61].…”

mentioning

confidence: 99%

The(q,t)-Gaussian process

Blitvić

2012

Journal of Functional Analysis

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The (q, t)-Fock space F q,t (H ), introduced in this paper, is a deformation of the q-Fock space of Bożejko and Speicher. The corresponding creation and annihilation operators now satisfy the commutation relationa defining relation of the Chakrabarti-Jagannathan deformed quantum oscillator algebra. The moments of the deformed Gaussian element s q,t (h) := a q,t (h) + a q,t (h) * are encoded by the joint statistics of crossings and nestings in pair partitions. The q = 0 < t specialization yields a natural single-parameter deformation of the full Boltzmann Fock space of free probability, with the corresponding semicircular measure variously encoded via the Rogers-Ramanujan continued fraction, the t-Airy function, the t-Catalan numbers of Carlitz-Riordan, and the first-order statistics of the reduced Wigner process.

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Factoriality of q-Gaussian von Neumann Algebras

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

Cited by 54 publications

References 5 publications

Fock space associated to Coxeter groups of type B

Fock space associated to Coxeter groups of type B

q-distributions in complex systems: a brief review

The(q,t)-Gaussian process

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