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162 product systems of von Neumann modules. Like in the proof for Arveson systems the crucial object is the generator of the associtated CPD-semigroup, there just a semigroup of positive definite Câvalued kernels with a conditionally positive definite kernel as generator. Here the situation is considerably more involved. The generator is a conditionally completely positive definite kernel and we are able to show that it has Christenson-Evans form (as conjectured in Theorem 5.4.14) only after having shown first that the GNS-system of a CP-semigroup (i.e. a product system generated by a single unit, a trivial thing in the case of Arveson systems) consists of time ordered Fock modules. Among other results which we have to provide before we can show this, there is a characterization of the endomporphisms of time ordered product systems (utilizing the elegant and purely algebraic ideas from Bhat [Bha99]) which allows us to find a powerful criterion to decide, whether a certain subset of units for a time ordered system is generating or not. En passant we show also that the results by Christenson and Evans [CE79] about the generator of a CP-semigroup are equivalent to the existence of a central unit (a unit consisting of centered elements) in a product system of von Neumann modules which has at least one continuous unit. In Chapter 14 we present two alternative constructions of product systems. The first one in Section 14.1 starts like Arveson from a (strict) E0 âsemigroup but on Ba (E) for some Hilbert Bâmodule (with a unit vector). This is a direct generalization to Hilbert modules of Bhatâs [Bha96] approach to Arveson systems. The second construction Section 14.2 is a simple generalization of the construction starting from a CP-semigroup to a construction starting from a system of transition expectations. In discrete time transition expectations are related to quantum Markov chains in the sense of Accardi [Acc74, Acc75]. The continuous time version is a generalization of a proposal by Liebscher [Lie00b] which we considered in Liebscher and Skeide [LS00b].
