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 von Neumann Modules versus Concrete von Neumann Modules

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

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

Ideal Submodules versus Ternary Ideals versus Linking Ideals

We show that ideal submodules and closed ternary ideals in Hilbert modules are the same; this contradicts the result of [Kol17]. We use this insight as a little peg on which to hang a little note about interrelations with other notions regarding Hilbert modules. In Section 1, we show (to our knowledge for the first time) that the ternary ideals (and equivalent notions) merit fully, in terms of homomorphisms and quotients, to be called ideals of (not necessarily full) Hilbert modules. The properties to be checked are intrinsically formulated for the modules (without any reference to the algebra over which they are modules) in terms of their ternary structure. The proofs, instead, are motivated from a third equivalent notion, linking ideals (Section 0), and a Theorem (Section 1) that all extends nicely to (reduced) linking algebras. As an application, in Section 2, we reprove most of the basic statements about extensions of Hilbert modules, by reducing their proof to the well-know analogue theorems about extensions of C*-algebras. Finally, in Section 3, we propose several new open problems that our method naturally suggests.Comment: 24 pages; corrected typos and minor cosmetics; new abstract; details added to the proof of Theorem 2.

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