CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond

These notes are the output of a decade of research on how the results about dilations of one-parameter CP-semigroups with the help of product systems, can be put forward to d–parameter semigroups –and beyond. While preliminary work on the two- and d–parameter case is based on the approach via the Arveson-Stinespring correspondence of a CP-map by Muhly and Solel [2002] (and limited to von Neumann algebras), here we explore consequently the approach via Paschke’s GNS correspondence of a CP-map [1973] by Bhat and Skeide [2000]. (A comparison is postponed to Appendix A (iv).) The generalizations are multi-fold, the difficulties often enormous. In fact, our only true if-and only- if theorem, is the following: A Markov semigroup over (the opposite of) an Ore monoid admits a full (strict or normal) dilation if and only if its GNS-subproduct system embeds into a product system. Already earlier, it has been observed that the GNS-(respectively, the Arveson Stinespring) correspondences form a subproduct system, and that the main difficulty is to embed that into a product system. Here we add, that every dilation comes along with a superproduct system (a product system if the dilation is full). The latter may or may not contain the GNS-subproduct system; it does, if the dilation is strong – but not only. Apart from the many positive results pushing forward the theory to large extent, we provide plenty of counter examples for almost every desirable statement we could not prove. Still, a small number of open problems remains. The most prominent: Does there exist a CP-semigroup that admits a dilation, but no strong dilation? Another one: Does there exist a Markov semigroup that admits a (necessarily strong) dilation, but no full dilation?