Deformation of independences in non-commutative probability spaces

非交换概率空间中独立性的变形

• 批准号: 14540201
• 负责人: YOSHIDA Hiroaki
• 金额: $ 1.47万
• 依托单位: Ochanomizu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540201/
• 关键词: operator algebras; non-commutative probability; quantum probability; deformation quantization; deformed convolutions; 量子碓率論

In usual probability space, if the pair of an algebra of bounded random variable on it and an expectation map then we can reconstruct the original probability space from such a pair of an algebra and an expectation map. The above algebra is commutative, hence, the usual probability space can be associated with a commutative algebra. Non-commutative probability space can be obtained by malting the algebra be non-commutative. Sometime such a procedure would be called quantization. Although the independence on usual probability spaces can be extended to a non-commutative probability space, it will require that independent random variables should be commutative. Unfortunately, this extension will not reflect well non-commutativity because the usual independence is based on tensor product. Voiculescu introduced the free independence which is based on free products and reflects well non-commutativity.If we are restricted that the independence should give the rule of calculation for mixed mom … More ents then only three kinds of independence (usual, free, and Boolean) are allowed in non-commutative probability space under some axioms. This is most explicit formalization of independence in non-commutative probability space. In general, independences should determine convolutions, and convolutions would give the moments-cumulants formulae. Standing this point of view, we can consider a more implicit deformed independence by deformations of moments-cumulants formula.In this project, we have adopted this procedure, that is, we have considered the deformation of independence by making deformations of moments-cumulants formulae. We have made several deformed free convolution, which interpolate free and Boolean convolutions. For the s-free and the r-free deformations, we investigated the corresponding Gaussian and Poisson random variables, especially on the s-free case, we have constructed the s-free Fock space (one of deformations of full Fock space) and gave the s-free Gaussian and the s-free Poisson random variables by the annihilation and the creation operators. Furthermore, we have extended the q-deformation, which is well known example that interpolates usual (the Boson Fock space) and Boolean (the Fermionic Fock space) convolutions, to 2-parameters cases. We call such a deformation the generalized q-deformation. As the generalized q-deformation, the (q,t) and the (q,s) deformations have been investigated and corresponding set partition statistics are also studied. Much more general deformed free convolution, the Delta-deformation, was introduced by Bozejko. We also succeeded to construct the weight function on non-crossing partitions for any given Delta convolution, which suggests us some kinds of new set partition statistics on non-crossing partitions. Less

在通常的概率空间中，如果有界随机变量的代数和期望映射的配对，则可以由这对代数和期望映射重构原始的概率空间。上述代数是可交换的，因此，通常的概率空间可以与交换代数相联系。非对易概率空间可以通过将代数转化为非对易而得到。有时，这样的过程将被称为量化。虽然对通常概率空间的独立性可以推广到非交换概率空间，但这将要求独立随机变量必须是可交换的。遗憾的是，这种推广不能很好地反映非对易性，因为通常的独立性是基于张量积的。Vocerescu提出了自由独立性，它建立在自由积的基础上，很好地反映了非交换性，如果对独立性进行限制，就应该给出混合MoM…的计算规则在一些公理下，非对易概率空间中只允许三种独立性(通常的、自由的和布尔的)。这是非对易概率空间中独立性的最明确的形式化。一般来说，独立性应该决定卷积，而卷积将给出矩-累积量公式。站在这个角度，我们可以考虑一种更隐式的通过矩-累积量公式变形的独立性。在这个项目中，我们采用了这个过程，即我们通过使矩-累积量公式变形来考虑独立性的变形。我们做了几个变形的自由卷积，它们插值自由卷积和布尔卷积。对于S自由和r自由变形，我们研究了相应的高斯和泊松随机变量，特别是在S自由的情况下，我们构造了S自由Fock空间(全Fock空间的变形之一)，并通过湮灭和创造算符给出了S自由的高斯和S自由的泊松随机变量。此外，我们还将通常的(玻色子-Fock空间)和布尔(费米子-Fock空间)卷积内插的著名例子Q-形变推广到双参数情形。我们称这种变形为广义Q-变形。作为广义q-变形，我们研究了(q，t)和(q，S)变形，并研究了相应的集合划分统计量。Bozejko提出了更一般的变形自由卷积--Delta变形。对于任意给定的Delta卷积，我们还成功地构造了非交叉划分的权函数，这给出了一些新的关于非交叉划分的集合划分统计量。较少

项目成果

Reiko Hori, Hiroaki Yoshida: "The spectrum radii of free convex sums of projections"Natural Science Report of Ochanomizu University. 54・2. 1-9 (2003)

堀丽子、吉田弘明：“自由凸投影和的谱半径”御茶水大学自然科学报告54・2（2003年）。

Remarks on the s-free convolution

关于s-free卷积的备注

• 发表时间: 2002
• 期刊: Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroad, QP-PQ 16
• 作者: A.Krystek;H.Yoshida;H.Yoshida;H.Yoshida
• 通讯作者: H.Yoshida

On a generalized arc-sine law for one-dimensional diffusion processes.

一维扩散过程的广义反正弦定律。

• 发表时间: 2005
• 期刊: Osaka J.Math. 42
• 作者: Kasahara;Y.;Yano Y.
• 通讯作者: Yano Y.

Anna Krystek, Hiroaki Yoshida: "The combinatorics of the r-free convolution"Infinite Dimensional Analysis, Quantum Probability and Related Topics. 6・4. 619-627 (2003)

Anna Krystek、Hiroaki Yoshida：“无 r 卷积的组合学”无限维分析、量子概率和相关主题 6・4（2003 年）。

Yuji Kasahara, Yuko Yano: "On a generalized arc-sine law for one-dimensional diffusion processes"Osaka J.Math.. (掲載予定).

Yuji Kasahara、Yuko Yano：“关于一维扩散过程的广义反正弦定律”Osaka J.Math..（待出版）。