论文标题
CP - 序列和扩张，亚产品系统和超产品系统：多参数案例及以后
CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond
论文作者
论文摘要
这些注释是关于如何借助产品系统的一参参数CP-序列的扩张结果的研究，可以提出D-参数半元素及以后的结果。在对两参数和D-参数案例上的工作是基于通过Muhly和Solel对CP-MAP的Arveson-Stinespring对应关系的方法（仅限于Von Neumann代数）的方法，因此，我们通过Bhat and Skeide和Skeide和Skeide探索了Paschke的GNS gns gns-gns-gns-gns-gns-gns-gns-gns-gns-gns-gns。 （将比较推迟到附录A（IV）。 概括是多重的，困难通常很大。实际上，如果定理，我们唯一的真实是：Markov Semigroup（与矿石相反）的Marodoid（相反）在且仅当其GNS-Subpropoduct System嵌入产品系统中时完全（严格或正常）的扩张。已经观察到的是，GNS-（分别是Arveson-Stinespring）的对应关系形成了亚产品系统，并且主要困难是将其嵌入到产品系统中。在这里，我们补充说，每个扩张都与超产品系统一起（如果扩张已满，则产品系统）。后者可能包含或可能不包含GNS-Subproduct System；如果扩张很强，但不仅如此。 除了在很大程度上推动理论的许多积极结果外，我们还为几乎所有无法证明的理想陈述提供了很多反示例。尽管如此，仍然存在少量的开放问题。最突出的是：是否存在承认扩张但没有强大扩张的CP - 序列？另一个：是否存在马尔可夫半群（Markov Semigroup）承认（一定是强大的）扩张，但没有充分的扩张？
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?
