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

摘要：本文研究了p可积函数的非交换空间的以下几个方面。如果p = 1，这样的空间就是von Neumann代数的前元，并且反映了底层算子代数的重要性质。回顾一下，冯诺依曼的前公数是否在迹类算子空间中有限表示仍然是开放的。在这里，我们将重点放在嵌入von Neumann代数前元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上。后者的研究使用了基于Pisier和Xu最近进展的鞅不等式。我们感兴趣的是非交换版本的Rosenthal/Burkholder不等式和Doob的极大不等式。极大不等式在（随机）分析中也是一个有用的工具。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。非交换概率为量子力学中的概率观点提供了一个可能的框架。这个理论结合了代数本质的基本概念与分析的洞察力和方法与微积分的根。p可积函数空间的非交换模拟在算子代数理论中有着悠久的传统，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与非交换理论之间的本质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的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