Conditional expectation and stochastic integrals in non-commutative Lp spaces

Abstract

The aim of the paper is to propose a general scheme for the consideration of non-commutative stochastic integrals. The role of a probability space is played by a couple (, φ0), where is a von Neumann algebra and φ0 is a faithful normal state on . Our processes live in the algebra of all measurable operators associated with the crossed product of by the modular automorphism group The algebra contains all the (Haagerup's) Lp spaces over . The measure topology of the algebra has the nice feature of inducing the Lp norm topology on the Lp spaces, which makes it particularly suitable for defining stochastic integrals. The commutative theory fits smoothly into the scheme, although there exists no canonical way of embedding the algebra of (commutative) random variables into . In fact, for any commutative stochastic process we have a family of different non-commutative stochastic processes corresponding to the process. This arbitrariness seems to be quite natural in the non-commutative context. An appropriate example can be found at the end of the paper (Section 6, C4).