CP-半群与 dilation、子积系统及超积系统：多参数情形及其他
CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond
AI总结：
本文将单参数CP-半群dilation的积系统方法推广到多参数情形，通过Paschke GNS对应展开研究，给出Ore幺半群上马尔可夫半群全dilation的充要条件，补充dilation与超积系统的关系，并提出两个突出开放问题。
摘要：
这些笔记是十年研究的成果，旨在探讨如何将借助积系统得到的单参数CP-半群的dilation结果推广到d参数半群乃至更一般的情形。现有关于双参数和d参数情形的工作基于Muhly和Solel通过CP-映射的Arveson-Stinespring对应（且限于冯·诺依曼代数）的方法，而本文则通过Bhat和Skeide提出的CP-映射的Paschke GNS对应来系统地探索这一问题（对比分析推迟至附录A(iv)）。推广是多方面的，困难也往往极大。事实上，我们唯一真正的充要定理如下：Ore幺半群（的反）上的马尔可夫半群存在全（严格或正规）dilation当且仅当其GNS-子积系统嵌入到一个积系统中。此前已有研究发现，GNS-（或Arveson-Stinespring-）对应构成子积系统，而主要困难在于将其嵌入积系统。本文补充：每个dilation都伴随一个超积系统（若dilation是全的，则为积系统），后者可能包含也可能不包含GNS-子积系统；当dilation是强的时包含，但不仅限于此。除了大量推动该理论发展的正面结果外，我们还为几乎所有未能证明的理想命题提供了丰富的反例。不过，仍有少量开放问题存在，最突出的是：是否存在允许dilation但不允许强dilation的CP-半群？另一个是：是否存在允许（必然是强的）dilation但不允许全dilation的马尔可夫半群？
英文摘要：
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?
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
