Deformation of independences in non-commutative probability spaces

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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