Independence and Product Systems

Starting from elementary considerations about independence and Markov processes in classical probability we arrive at the new concept of conditional monotone independence (or operator-valued monotone independence). With the help of product systems of Hilbert modules we show that monotone conditional independence arises naturally in dilation theory. One of the most fundamental concepts in probability theory — in the sequel, we say classical probability — is independence. The most fundamental concept dealing with non-independent random variables is the Markov property . In order to underline the formal aspect of the Markov property being a generalization of independence (replacing expectations by conditional expectations) we will say conditional independence (which operator algebraists would prefer to call operatorvalued independence). The expectation is a normalized positive linear functional (i.e. a state) on the commutative algebra of random variables. Noncommutative or quantum probability is a noncommutative generalization of classical probability designed to include also quantum physical applications. A quantum probability space is, therefore, a pair (A, φ) of a unital ∗–algebra (more specifically, a (pre– )C–algebra or a von Neumann algebra) A with a normalized (i.e. φ(1) = 1) positive (i.e. φ(aa) ≥ 0) linear functional φ : A → C. In this context a classical probability space (Ω,F, P ) corresponds to the commutative quantum probability space (L(Ω), φ = ∫ • dP ). Classical independence is symmetric, i.e. if X1 is independent of X2 (conditioning X1 on X2 does not change probabilities of X1), then X2 is independent of X1, too. In Section 1 we discuss an example where this symmetry is not desirable. However, within the category of commutative quantum probability spaces there is only one independence, namely, that one which gives back classical independence. Therefore, if we want to model the mentioned example, then we are forced to leave the category of commutative quantum probability spaces even for classical random variables. The only noncommutative independence which earns the name independence and is not symmetric is monotone independence ; see Section 1. In Section 2 we discuss the case of conditional independence, which is the basis for our discussion of Markov processes, later on. As a new concept we introduce conditional monotone independence (or operator-valued monotone independence). As long as we are dealing with classical random variables, both conditional independence and conditional monotone independence are possible. However, if we 1