Direct producted $W^{*}$-probability spaces and corresponding amalgamated free stochastic integration

Bull. Korean Math. Soc. 2007 Vol. 44, No. 1, 131-150 Printed March 1, 2007

Ilwoo Cho Saint Ambrose University

Abstract : In this paper, we will define direct producted $W^{*}$-probability spaces over their diagonal subalgebras and observe the amalgamated freeness on them. Also, we will consider the amalgamated free stochastic calculus on such free probabilistic structure. Let $(A_{j},$ $\varphi _{j})$ be a tracial $W^{*}$-probability spaces, for $j$ $=$ $1,\ldots,N.$ Then we can define the corresponding direct producted $W^{*}$-probability space $(A,$ $% E) $ over its $N$-th diagonal subalgebra $D_{N}$ $\equiv $ $\Bbb{C}^{\oplus N},$ where $A$ $=$ $\oplus _{j=1}^{N}$ $A_{j}$ and $E$ $=$ $\oplus _{j=1}^{N} $ $\varphi _{j}.$ 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 $D_{N}$-semicircularity 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 It\^{o}'s formula is naturally extended to the $D_{N}$-valued case.

Keywords : direct producted $W^{*}$-probability spaces over their diagonal subalgebras, $D_{N}$-freeness, $D_{N}$-semicircularity, $D_{N}$-valued infinitely divisibility, $D_{N}$-valued simple adapted biprocesses, $D_{N}$-valued free stochastic integrals, It\^{o}'s