Commutants of von Neumann Modules, Representations of B^a(E) and Other Topics Related to Product Systems of Hilbert Modules

Details Commutants of von Neumann Modules, Representations of B^a(E) and Other Topics Related to Product Systems of Hilbert Modules Commutants of von Neumann Modules, Representations of B^a(E) and Other Topics Related to Product Systems of Hilbert Modules

2003-08-25

arxiv.org/abs/math/0308231v1

In G.L. Price, B.M. Baker, P.E.T. Jorgensen, and P.S. Muhly, editors, Advances in quantum dynamics, number 335 in Contemporary

Mathematics

Operator Algebras

To appear in Proceedings of ``Advances in Quantum Dynamics'', Mount Holyoke, 2002

Scientific paper

We review some of our results from the theory of product systems of Hilbert modules. We explain that the product systems obtained from a CP-semigroup in a paper by Bhat and Skeide and in a paper by Muhly and Solel are commutants of each other. Then we use this new commutant technique to construct product systems from E_0-semigroups on B^a(E) where E is a strongly full von Neumann module. (This improves the construction from a paper by Skeide for Hilbert modules where existence of a unit vector is required.) Finally, we point out that the Arveson system of a CP-semigroup constructed by Powers from two spatial E_0-semigroups is the product of the corresponding spatial Arveson systems as defined (for Hilbert modules) in a paper by Skeide. It need not coincide with the tensor product of Arveson systems.

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