Deformation of independences in non-commutative probability spaces

在通常的概率空间中，如果有一个有界随机变量的代数和一个期望映射对在其上，那么我们可以用这样一个代数和一个期望映射对重构原始概率空间。上述代数是交换的，因此，通常的概率空间可以与交换代数联系起来。非交换概率空间可以通过将代数定义为非交换得到。有时这样的过程被称为量子化。虽然在一般概率空间上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。不幸的是，这个扩展不能很好地反映非交换性，因为通常的独立性是基于张量积的。Voiculescu引入了自由独立性，它基于自由积，很好地反映了非交换性。在非交换概率空间中，如果我们被限制独立性必须给出混合矩阵的计算规则，那么在某些公理下，非交换概率空间只允许三种独立性（普通独立性、自由独立性和布尔独立性）。这是非交换概率空间中独立性的最显式形式化。一般来说，独立性应该决定卷积，卷积会给出矩累积量的公式。站在这个观点，我们可以考虑一个更隐式的变形独立性矩累积公式的变形。在这个项目中，我们采用了这个程序，即我们通过制作弯矩-累积量的变形公式来考虑独立变形。我们做了几个变形的自由卷积，其中插值自由和布尔卷积。对于s-free和r-free变形，我们研究了相应的高斯和泊松随机变量，特别是在s-free情况下，我们构造了s-free Fock空间（全Fock空间的一种变形），并通过湮灭算子和生成算子给出了s-free高斯和s-free泊松随机变量。此外，我们已经扩展了q-变形，这是众所周知的插值通常（玻色子Fock空间）和布尔（费米子Fock空间）卷积的例子，到2参数的情况。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，并研究了相应的集划分统计量。更一般的变形自由卷积，三角洲变形，是由Bozejko引入的。我们还成功地构造了任意给定的Delta卷积的非交叉分区上的权函数，这给我们提供了一些新的非交叉分区上的集分区统计量。少

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