Product Systems; a Survey with Commutants in View
arXiv:0709.0915 · doi:10.1142/9789812832962_0004
Abstract
The theory of product systems both of Hilbert spaces (Arveson systems) and product systems of Hilbert modules has reached a status where it seems appropriate to rest a moment and to have a look at what is known so far and what are open problems. However, the attempt to give an approximately complete account in view pages is destined to fail already for Arveson systems since Tsirelson, Powers and Liebscher have discovered their powerful methods to construct large classes of examples. In this survey we concentrate on that part of the theory that works also for Hilbert modules. This does not only help to make a selection among the possible topics, but it also helps to shed some new light on the case of Arveson systems. Often, proofs that work for Hilbert modules also lead to simpler proofs in the case of Hilbert spaces. We put emphasis on those aspects that arise from recent results about commutants of von Neumann correspondences, which, in the case of Hilbert spaces, explain the relation between the Arveson system and the Bhat system associated with an E_0--semigroup on B(H).
Invited survey, to appear in the proceedings of the 2006 Quantum Probability Conference in Nottingham
References in corpus (9)
- From random sets to continuous tensor products: answers to three questions of W. Arveson
- Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces
- Commutants of von Neumann Modules, Representations of B^a(E) and Other Topics Related to Product Systems of Hilbert Modules
- From slightly coloured noises to unitless product systems
- Nondegenerate Representations of Continuous Product Systems
- Unit Vectors, Morita Equivalence and Endomorphisms
- On the existence of -semigroups
- Quantum Markov Semigroups (Product Systems and Subordination)
- On the universal property of Pimsner-Toeplitz C*-algebras and their continuous analogues