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

TL;DR

This work develops a comprehensive dilation theory for CP-semigroups indexed by multi-parameter monoids, extending the classical one-parameter framework. Central to the approach are GNS-subproduct systems, subproduct and superproduct systems, and the construction of dilations that yield E-semigroups; crucial results show that a Markov semigroup over the opposite of an Ore monoid has a full dilation iff its GNS-subproduct system embeds into a product system. The notes unify discrete and continuous time settings, introduce exponentiation to pass from discrete to continuous multi-parameter systems, and analyze both positive results and counterexamples, including cases where dilations fail to exist or are not “good.” They also connect to unitalization, Morita equivalence, and left dilations, providing a toolbox for tackling multi-parameter quantum dynamics and their dilations across C*-algebras and von Neumann algebras. The work culminates in several open questions, highlighting subtle boundaries between embeddability, strong/dilation notions, and the role of topology in the existence and structure of dilations. Overall, it offers a robust framework for constructing, classifying, and understanding multi-parameter dilations and their associated product-system structures, with implications for strongly commuting CP-semigroups and quantum dynamical semigroups.

Abstract

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 exisiting 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 (and limited to von Neumann algebras), here we explore consequently the approach via Paschke's GNS-correspondence of a CP-map by Bhat and Skeide. (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?