Type I product systems of Hilbert modules

Abstract

Christensen and Evans showed that, in the language of Hilbert modules, a bounded derivation on a von Neumann algebra with values in a two-sided von Neumann module (i.e. a sufficiently closed two-sided Hilbert module) are inner. Then they use this result to show that the generator of a normal uniformly continuous completely positive (CP-) semigroup on a von Neumann algebra decomposes into a (suitably normalized) CP-part and a derivation like part. The backwards implication is left open. In these notes we show that both statements are equivalent among themselves and equivalent to a third one, namely, that any type I tensor product system of von Neumann modules has a unital central unit. From existence of a central unit we deduce that each such product system is isomorphic to a product system of time ordered Fock modules. We, thus, find the analogue of Arveson's result that type I product systems of Hilbert spaces are symmetric Fock spaces. On the way to our results we have to develop a number of tools interesting on their own right. Inspired by a very similar notion due to Accardi and Kozyrev, we introduce the notion of semigroups of completely positive definite kernels (CPD-semigroups), being a generalization of both CP-semigroups and Schur semigroups of positive definite C -valued kernels. The structure of a type I system is determined completely by its associated CPD-semigroup and the generator of the CPD-semigroup replaces Arveson's covariance function. As another tool we give a complete characterization of morphisms among product systems of time ordered Fock modules. In particular, the concrete form of the projection endomorphisms allows us to show that subsystems of time ordered systems are again time ordered systems and to find a necessary and sufficient criterion when a given set of units generates the whole system. As a byproduct we find a couple of characterizations of other subclasses of morphisms. We show that the set of contractive positive endomorphisms are order isomorphic to the set of CPD-semigroups dominated by the CPD-semigroup associated with type I system.