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

摘要本文的目的是研究p-可积函数的非交换空间的以下几个方面。如果p为1，则这样的空间是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，von Neumann的前项是否有限地表示在迹类算子空间中仍然是开放的。在这里，我们集中讨论了嵌入到von Neumann代数的准对偶中的有限维空间的等距刻画及其与李代数和(非交换)随机过程理论的联系。后者的研究基于Pisier和Xu的最新进展，使用了鞅不等式。我们对非对易形式的Rosenthal/Burkhold不等式和Doob的极大值不等式感兴趣。极大不等式也被认为是(随机)分析中的一个有用工具。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种可能的框架。这一理论将代数性质的基本概念与分析洞察力相结合，将方法与微积分中的根结合起来。P-可积函数空间的非对易模拟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卓有成效的框架。揭示或克服对易理论与非对易理论之间的本质差异是最具挑战性的。这一领域使数学界和数学物理学内部的不同流派之间能够相互作用。这种互动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