DIRECT PRODUCTED W*-PROBABILITY SPACES AND CORRESPONDING AMALGAMATED FREE STOCHASTIC INTEGRATION

Title & Authors
DIRECT PRODUCTED W*-PROBABILITY SPACES AND CORRESPONDING AMALGAMATED FREE STOCHASTIC INTEGRATION
Cho, Il-Woo;

Abstract
In this paper, we will define direct producted $\small{W^*-porobability}$ spaces over their diagonal subalgebras and observe the amalgamated free-ness on them. Also, we will consider the amalgamated free stochastic calculus on such free probabilistic structure. Let ($\small{A_{j},\;{\varphi}_{j}}$) be a tracial $\small{W^*-porobability}$ spaces, for j = 1,..., N. Then we can define the corresponding direct producted $\small{W^*-porobability}$ space (A, E) over its N-th diagonal subalgebra $\small{D_{N}\;{\equiv}\;\mathbb{C}^{{\bigoplus}N}}$, where $\small{A={\bigoplus}^{N}_{j=1}\;A_{j}\;and\;E={\bigoplus}^{N}_{j=1}\;{\varphi}_{j}}$. In Chapter 1, we show that $\small{D_{N}-valued}$ cumulants are direct sum of scalar-valued cumulants. This says that, roughly speaking, the $\small{D_{N}-freeness}$ is characterized by the direct sum of scalar-valued freeness. As application, the $\small{D_{N}-semicircularityrity}$ and the $\small{D_{N}-valued}$ infinitely divisibility are characterized by the direct sum of semicircularity and the direct sum of infinitely divisibility, respectively. In Chapter 2, we will define the $\small{D_{N}-valued}$ stochastic integral of $\small{D_{N}-valued}$ simple adapted biprocesses with respect to a fixed $\small{D_{N}-valued}$ infinitely divisible element which is a $\small{D_{N}-free}$ stochastic process. We can see that the free stochastic Ito's formula is naturally extended to the $\small{D_{N}-valued}$ case.
Keywords
direct producted $\small{W^*-probability}$ spaces over their diagonal subalgebras;$\small{D_N}$-freeness;$\small{D_N}$-semicircularity;$\small{D_N}$-valued infinitely divisibility;$\small{D_N}$-valued simple adapted biprocesses;$\small{D_N}$-valued free stochastic integrals;$\small{It\^{o}$ formula;$\small{D_N}$-free brownian motions;
Language
English
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