Abstract
In this paper, we will define direct producted $W^*-porobability$ 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 ($A_{j},\;{\varphi}_{j}$) be a tracial $W^*-porobability$ spaces, for j = 1,..., N. Then we can define the corresponding direct producted $W^*-porobability$ space (A, E) over its N-th diagonal subalgebra $D_{N}\;{\equiv}\;\mathbb{C}^{{\bigoplus}N}$, where $A={\bigoplus}^{N}_{j=1}\;A_{j}\;and\;E={\bigoplus}^{N}_{j=1}\;{\varphi}_{j}$. In Chapter 1, we show that $D_{N}-valued$ cumulants are direct sum of scalar-valued cumulants. This says that, roughly speaking, the $D_{N}-freeness$ is characterized by the direct sum of scalar-valued freeness. As application, the $D_{N}-semicircularityrity$ and the $D_{N}-valued$ 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 $D_{N}-valued$ stochastic integral of $D_{N}-valued$ simple adapted biprocesses with respect to a fixed $D_{N}-valued$ infinitely divisible element which is a $D_{N}-free$ stochastic process. We can see that the free stochastic Ito's formula is naturally extended to the $D_{N}-valued$ case.