Non-commutative Lp-spaces and their Connection to Probability and Operator Spaces

非交换 Lp 空间及其与概率和算子空间的联系

基本信息

批准号：
0088928
负责人：
Marius Junge
金额：
$ 8.77万
依托单位：
University of Illinois at Urbana-Champaign
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2000
资助国家：
美国
起止时间：
2000-08-01 至 2004-01-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0088928&HistoricalAwards=false
关键词：
Non commutative Lp spaces their

项目摘要

AbstractJungeThe aim of this research is the investigation of the followingdifferent aspects of non-commutative spaces of p-integrable functions. If p is 1 such a space is the predual of von Neumann algebra and reflects important properties of the underlying operator algebra. We recall, that it is still open whether preduals of von Neumann are finitely represented in the space of trace class operators. Here, we focus on isometric characterization of finite dimensional spaces embedding into the predual of a von Neumann algebra and its connection to the theory of Lie-algebras and (non-commutative) stochastical processes. The investigation of the latter uses martingale inequalities based on recent progress by Pisier and Xu. We are interested in the non-commutative version of the Rosenthal/Burkholder inequality and Doob's maximal inequality. Maximal inequalities are also known as a useful tool in (stochastical) analysis. The more recent theory of operator spaces delivers the right framework for these investigations and reveals surprising properties of the non-commutative space of p-integrable functions associated to free groups. Non-commutative probability provides one possible framework for the probabilistic viewpoint in quantum mechanics. This theorycombines fundamental concepts of algebraic nature with analyticinsight and methods with roots in calculus. The non-commutative analogue for the spaces of p-integrable functions has a long tradition in the theory of operator algebras and provides a fruitful framework for understanding classical tools in probability. It is most challenging to reveal or overcome substantial differences between the commutative and non-commutative theory. This area enables the interaction between different streams inside the mathematical community and mathematical physics. This kind of interaction is one of the most important resources for new development in mathematics

摘要Junge 这项研究的目的是研究 p-可积函数非交换空间的以下不同方面。如果 p 为 1，则这样的空间是冯·诺依曼代数的预置空间，并且反映了基础算子代数的重要属性。我们记得，冯·诺依曼的预分项是否在迹类算子的空间中有限地表示仍然是一个悬而未决的问题。在这里，我们重点关注嵌入冯·诺依曼代数预演的有限维空间的等距表征及其与李代数和（非交换）随机过程理论的联系。后者的研究使用基于 Pisier 和 Xu 最近进展的鞅不等式。我们对罗森塔尔/伯克霍尔德不等式和杜布最大不等式的非交换版本感兴趣。最大不平等也被认为是（随机）分析中的有用工具。最新的算子空间理论为这些研究提供了正确的框架，并揭示了与自由群相关的 p 可积函数的非交换空间的令人惊讶的性质。非交换概率为量子力学中的概率观点提供了一种可能的框架。该理论将代数性质的基本概念与分析洞察力和源于微积分的方法结合起来。 p-可积函数空间的非交换类比在算子代数理论中有着悠久的传统，并为理解概率中的经典工具提供了一个富有成效的框架。揭示或克服交换理论和非交换理论之间的实质性差异是最具挑战性的。该区域使得数学界和数学物理内部不同流派之间能够进行互动。这种互动是数学新发展最重要的资源之一