CPSemigroups and Dilations, Subproduct Systems and Superproduct Systems: The MultiParameter Case and Beyond
Abstract
These notes are the output of a decade of research on how the results about dilations of oneparameter CPsemigroups with the help of product systems, can be put forward to dparameter semigroups  and beyond. While preliminary work on the two and dparameter case is based on the approach via the ArvesonStinespring correspondence of a CPmap by Muhly and Solel (and limited to von Neumann algebras), here we explore consequently the approach via Paschke's GNScorrespondence of a CPmap by Bhat and Skeide. (A comparison is postponed to Appendix A(iv).) The generalizations are multifold, the difficulties often enormous. In fact, our only true ifandonlyif 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 GNSsubproduct system embeds into a product system. Already earlier, it has been observed that the GNS (respectively, the ArvesonStinespring) 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 GNSsubproduct 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 CPsemigroup 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?
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.05166
 Bibcode:
 2020arXiv200305166S
 Keywords:

 Mathematics  Operator Algebras;
 46L55;
 46L07;
 46L53
 EPrint:
 219 pages. (We might opt to add an index and/or a "road map" in later version.)