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

Abstract

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