Product systems of Hilbert modules and their applications in quantum dynamics

Markov semigroups (semigroups of unital completely positive maps), E0{ semigroups (semigroups of unital endomorphisms), and dilations of the for- mer to the latter, are intimately related to and connected by product systems. This has been illustrated in the lecture of Rajarama Bhat for the case when the semigroups act on B(H), the algebra of all bounded linear operators on a Hilbert space H. When the Markov semigroups act on a unital C{algebra B, the dilating E0{semigroups will act on the algebra Ba(E) of all adjointable operators on a Hilbert B{module E, and the related product systems consist of correspon- dences over B. We immediately see a rst dierence that illustrates how much more ex- ible this setting is, when we look at the concept of units. Like in the Hilbert space case, units help classifying product systems. But unlike units in Arve- son systems (that is, product systems of Hilbert spaces), units in product sys- tems are related to CP-semigroups (semigroups of completely positive maps) or, more generally, to CPD-semigroups (semigroups of completely positive denite kernels). A single unit can generate an interesting product system. The units that behave most similar to (normalised) units in Arveson systems are central (unital) units (that is, units whose elements commute with the elements of the algebra B). A product system that admits a central unital unit is called spatial. Spatiality of a product systems corresponds to spatiality of the E0-semigroup from which it is derived. On the level of CP- semigroups the same statement is true only for von Neumann algebras B (and von Neumann modules). For unital C{algebras equivalence of spatiality of a CP-semigroup to that the product system of that CP-semigroup embeds into a spatial one. Spatiality of both the dynamics and the associated product system of the dynamics is a key property for this school! In fact, when a product system is spatial it is easy to construct a so-called noise having that product system. Noises come along with monotone independent ltrations. (In the B(H){case also tensor independent.) A monotone noise may always be \blown up" to a free noise. If the product system of a noise is also the product system of an- other E0{semigroup, then that E0{semigroup and the noise are stably cocycle conjugate (that is, suitable amplications of them are cocycle conjugate). As a consequence, a Markov semigroup allows a Hudson-Parthasarathy dilation (that is, more or less a unital dilation which is a cocycle perturbation of a noise) if and only if it is spatial. This subsummarises all existence results obtained by quantum stochastic calculus. However, this abstract construc- tion does not address the problem if the perturbing cocycle is adapted to a ltration or if it fulls a quantum stochastic dierential equation. But, it tells us that we need not look for a Hudson-Parthasarathy dilation if the Markov semigroup is not spatial. We give a preliminary outline of our ve lectures. I. Product systems and E0{semigroups: Although in this school we are mainly interested in product systems of E0{semigroups that dilate some Markov semigroup, specically in spatial product systems, for didactic reasons it is convenient to look rst at the general relation between product systems and arbitrary E0-semigroups. In this section we dis- cuss the one-to-one correspondence between product systems (up to isomorphism) and E0{semigroups (up to stable cocycle conjugacy). II. Product systems and CP(D)-semigroups: Every CP-semigroup has a product system. If the semigroup is even Markov, then we easily con- struct a dilation with help of that product system. We do a similar thing also for CPD-semigroups. III. Spatial product systems: In this lecture we present the basic classica- tion of product systems into spatial and nonspatial, and into types I, II, III. We show that spatial type I systems are Fock, and that type I sys- tems of von Neumann correspondences are spatial automatically. We dene an index for spatial product systems and show that this index is additive under a suitable product of spatial product systems. IV. Spatial dynamics and noises: In this lecture we discuss spatial dilations of spatial Markov semigroups by cocycle perturbations of noises. We brie y sketch the relation with free product systems and free noises. V. Commutants of von Neumann correspondences and representations of product systems: Actually, Arveson's original approach to Arveson sys- tems from E0{semigroups on B(H) is dual to Bhat's approach, which is the basis for our discussion of the general case. (In fact, the two Arveson systems are anti-isomorphic and need not be isomorphic.) In the general context, the duality is described as the commutant on the level of product systems of von Neumann correspondences.