Classification of E_0-semigroups by product systems

In his Memoir from 1989, Arveson started the modern theory of product systems. More precisely, with each E0-semigroup (that is, a unital endomorphism semigroup) on B(H) he associated a product system of Hilbert spaces (Arveson system, henceforth). He also showed that the Arveson system determines the E0- semigroup up to cocycle conjugacy. In three successor papers, Arveson showed that every Arveson system comes from an E0-semigroup. There is, therefore, a one-toone correspondence between E0-semigroups on B(H) (up to cocycle conjugacy) and Arveson systems (up to isomorphism). In the meantime, product systems of correspondences (or Hilbert bimodules) have been constructed from Markov semigroups on general unital C.-algebras or on von Neumann algebras. These product systems showed to be an efficient tool in the construction of dilations of Markov semigroups to E0-semigroups and to automorphism groups. In particular, product systems over correspondences over commutative algebras (as they arise from classical Markov processes) or other algebras with nontrivial center, show surprising features that can never happen with Arveson systems. A dilation of a Markov semigroup constructed with the help of a product system always acts on Ba(E), the algebra of adjointable operators on a Hilbert module E. (If the Markov semigroup is on B(H) then E is a Hilbert space.) Only very recently, we showed that every product system can occur as the product system of a dilation of a nontrivial Markov semigroup. This makes it necessary to extend the theory to the relation between E0-semigroups on Ba(E) and product systems of correspondences. In these notes we present a complete theory of classification of E0-semigroups by product systems of correspondences. As an application of our theory, we answer the fundamental question if a Markov semigroup admits a dilation by a cocycle perturbations of noise: It does if and only if it is spatial.