Classification of E0–Semigroups by Product Systems

In these notes we tie up some loose ends in the theory of E0–semigroups and their classification by product systems of Hilbert modules. We explain how the notion of cocycle conjugacy must be modified in order to see how product systems classify E0–semigroups. Actually, we will find two notions of cocycle conjugacy (which for Hilbert spaces coincide) that lead to classification up to isomorphism of product systems and up to Morita equivalence of product systems, respectively. (In between there is also a classification up to generalized isomorphism of product systems.) Apart from these new results, we provide also general versions of results known for Hilbert modules with unit vectors. In this context it is also indispensable to review the notions of Morita equivalent product systems and Morita equivalent Hilbert modules, adding some generalities that have not yet been mentioned. In any case, we underline the outstanding role played by Morita equivalence in the relation between E0–semigroups and product systems. As usual with Morita equivalence, the most satisfying form of the results we find for von Neumann algebras. Some of the C∗–versions of the results will depend on countability assumptions. Altogether, we have now a complete the theory of the classification of normal E0–semigroups on Ba(E) by product systems of von Neumann correspondences. We have the same theory for the classification of strict E0–semigroups by product systems of C∗–correspondences under countability hypotheses. In both cases, we apply our theory to prove that a Markov semigroup admits a Hudson-Parthasarathy dilation if and only if it is spatial. ∗This work is supported by research funds of University of Molise and Italian MIUR.