Free subproducts and free scaled products of II_1 factors

$C^*$-子积系统与积系统

$C^*$-subproduct and product systems

本文引入并研究了 $C^*$-代数的双参数子积系统与积系统，证明了子积系统可膨胀为积系统并组装为 $C^*$-代数，同时揭示了其余单位与幂等状态及 Tsirelson 子积系统的关联。

二次子乘积系统、自由积及其C*-代数

Quadratic subproduct systems, free products, and their C*-algebras

本文受二次代数等领域相互作用启发，引入希尔伯特空间的二次子乘积系统概念，研究其诱导系统及对应代数，定义子乘积系统范畴的自由积运算并揭示其与Toeplitz代数约化自由积的对应关系，还得到相关代数K-理论结果。

Product systems, subproduct systems and dilation theory of completely positive semigroups

This thesis is dedicated to developing a dilation theory for semigroups of completely positive maps. The first part treats two-parameter semigroups, and contains also contributions to dilation theory of product system representations. The second part deals with completely positive semigroups parameterized by quite general semigroups, where the major technical tool introduced is subproduct systems and their representations. In the third part subproduct systems are studied, together with the multivariable operator theory and operator algebras they give rise to.

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

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。

Subproduct systems

The notion of a subproduct system, a generalization of that of a product system, is introduced. We show that there is an essentially 1 to 1 correspondence between cp-semigroups and pairs (X,T) where X is a subproduct system and T is an injective subproduct system representation. A similar statement holds for subproduct systems and units of subproduct systems. This correspondence is used as a framework for developing a dilation theory for cp-semigroups. Results we obtain: (i) a *-automorphic dilation to semigroups of *-endomorphisms over quite general semigroups; (ii) necessary and sufficient c…

秩一乘积格的非交换拓扑因子定理

The noncommutative topological factor theorem for rank-one product lattices

本文证明实秩一单李群乘积中不可约格的非交换拓扑因子定理，其结论可推广至树及混合局部域版本，且$\text{SL}_3(\boldsymbol{Z})$全旗作用的对应分类将蕴含普通ITAP。