Tensor product systems of Hilbert modules and dilations of completely positive semigroups

Abstract

In this paper we study the problem of dilating unital completely positive (CP) semigroups (quantum dynamical semigroups) to weak Markov flows and then to semigroups of endomorphisms (E₀-semigroups) using the language of Hilbert modules. This is a very effective, representation free approach to dilation. In this way we are able to identify the right algebra (maximal in some sense) for endomorphisms to act. We are led inevitably to the notion of tensor product systems of Hilbert modules and units for them, generalizing Arveson's notions for Hilbert spaces. In the course of our investigations we are not only able to give new natural and transparent proofs of well-known facts for semigroups on B(H), but also extend the results immediately to much more general setups. For instance, Arveson classifies E₀-semigroups on B(H) up to cocycle conjugacy by product systems of Hilbert spaces. We find that conservative CP-semigroups on arbitrary unital C-algebras are classified up to cocycle conjugacy by product systems of Hilbert modules. Looking at other generalizations, it turns out that the role played by E₀-semigroups on B(H) in dilation theory for CP-semigroups on B(G) is now played by E₀-semigroups on B^a(E), the full algebra of adjointable operators on a Hilbert module E. We have CP-semigroup versions of many results proved by Paschke for CP maps.