E_0-Semigroups for Continuous Product Systems

We show that every continuous product system of correspondences over a unital C*-algebra occurs as the product system of a strictly continuous E_0-semigroup

Hilbert Modules - Square Roots of Positive Maps

We reflect on the notions of positivity and square roots. We review many examples which underline our thesis that square roots of positive maps related to *-algebras are Hilbert modules. As a result of our considerations we discuss requirements a notion of positivity on a *-algebra should fulfill and derive some basic consequences.Comment: 24 page

A Factorization Theorem for φ\varphi--Maps

We present a far reaching generalization of a factorization theorem by Bhat, Ramesh, and Sumesh (stated first by Asadi) and furnish a very quick proof.Comment: Reivised according to the referee's suggestions (now 5 pages); to appear in Journal of Operator Theor

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.Comment: To appear in Proceedings of the ``First Sino-German Meeting on Stochastic Analysis'', Beijing, 200

Hilbert von Neumann Modules versus Concrete von Neumann Modules

Hilbert von Neumann modules and concrete von Neumann modules are the same thing.Comment: 2 page

Isometric Dilations of Representations of Product Systems via Commutants

We construct a weak dilation of a not necessarily unital CP-semigroup to an E-semigroup acting on the adjointable operators of a Hilbert module with a unit vector. We construct the dilation in such a way that the dilating E-semigroup has a pre-assigned product system. Then, making use of the commutant of von Neumann correspondences, we apply the dilation theorem to proof that covariant representations of product systems admit isometric dilations.Comment: Switched some definitions directly after theorems in Sect. 1, included (>6 pp.) introduction to von Neumann correspondences (Sect. 3). To appear in International Journal of Mathematic

Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations

For many Markov semigroups dilations in the sense of Hudson and Parthasarathy, that is a dilation which is a cocycle perturbation of a noise, have been constructed with the help of quantum stochastic calculi. In these notes we show that every Markov semigroup on the algebra of all bounded operators on a separable Hilbert space that is spatial in the sense of Arveson, admits a Hudson-Parthasarathy dilation. In a sense, the opposite is also true. The proof is based on general results on the the relation between spatial E_0-semigroups and their product systems