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

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 44, Issue 1, 2007, pp.131-150
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2007.44.1.131

Title & Authors

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

Cho, Il-Woo;

Cho, Il-Woo;

Abstract

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.

Keywords

direct producted spaces over their diagonal subalgebras;-freeness;-semicircularity;-valued infinitely divisibility;-valued simple adapted biprocesses;-valued free stochastic integrals; formula;-free brownian motions;

Language

English

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