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

本文的目的是研究p-可积函数的非交换空间的以下几个方面。如果p是1，这样的空间是von Neumann代数的预对偶，并且反映了基础算子代数的重要性质。我们还记得，它仍然是开放的，是否冯诺依曼的预言是在迹类运算符空间中的非线性表示。在这里，我们专注于等距特征的有限维空间嵌入到predual的冯诺依曼代数及其连接到理论的李代数和（非交换）随机过程。后者的调查使用鞅不等式的基础上最近的进展皮西尔和徐。我们感兴趣的是Rosenthal/Burkholder不等式和Doob极大不等式的非交换形式。极大不等式也被认为是（随机）分析中的有用工具。最近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惊人性质。非对易概率为量子力学中的概率观点提供了一个可能的框架。这一理论结合了代数性质的基本概念与分析的洞察力和方法与根在微积分。p-可积函数空间的非交换类似物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。这个区域使数学社区和数学物理内部不同流之间的交互成为可能。这种相互作用是数学新发展的最重要资源之一

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