In this paper, we will define direct producted

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 (

) be a tracial

spaces, for j = 1,..., N. Then we can define the corresponding direct producted

space (A, E) over its N-th diagonal subalgebra

, where

. In Chapter 1, we show that

cumulants are direct sum of scalar-valued cumulants. This says that, roughly speaking, the

is characterized by the direct sum of scalar-valued freeness. As application, the

and the

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

stochastic integral of

simple adapted biprocesses with respect to a fixed

infinitely divisible element which is a

stochastic process. We can see that the free stochastic Ito's formula is naturally extended to the

case.