Summary: We define direct producted \(W^*\)-probability spaces over their diagonal subalgebras and observe the amalgamated freeness on them. Also, we consider the amalgamated free stochastic calculus on such free probabilistic structure. Let \((A_j, \varphi_j)\) be a tracial \(W^*\)-probability spaces, for \(j = 1,\dots,N\). Then we can define the corresponding direct producted \(W^*\)-probability space \((A, E)\) over its \(N\)-th diagonal subalgebra \(D_N\equiv\mathbb C^{\oplus N}\), 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 sums of scalar-valued cumulants. This says that, roughly speaking, the \(D_N\)-freeness is characterized by the direct sum of scalar-valued freeness. As an application, the \(D_N\)-semicircularity and the \(D_N\)-valued infinite divisibility are characterized by the direct sum of semicircularity and the direct sum of infinite divisibility, respectively.
In Chapter 2, we 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 see that the free stochastic Itô’s formula is naturally extended to the \(D_N\)-valued case.