Product Systems; a Survey with Commutants in View

TL;DR

This survey reviews the current state of product systems of Hilbert spaces and modules, emphasizing recent results on commutants and their implications for understanding Arveson systems and related structures.

Contribution

It highlights the aspects of product systems that extend to Hilbert modules and clarifies their relation to Arveson systems through commutant theory.

Findings

Recent results on commutants shed light on the structure of product systems.

Simpler proofs for Hilbert spaces are derived from Hilbert module techniques.

The survey identifies open problems and directions for future research.

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).