Lp Estimates in Non-commutative Probability and Analysis

“非对易概率与分析中的Lp估计”这个项目的出发点是Pisier/Xu，Randrianantoanina，Junge和Junge/Xu得到的非对易鞅不等式的最新结果和技巧。这一新的见解使我们能够证明一个非交换的类比，即完全正映射的极大遍历定理，并在此背景下研究平方函数不等式(与LeMerdy/Xu共同工作)。它的提法和性质是由算子空间理论驱动的。另一方面，对于理解非交换LP空间中独立的、不可分辨的、可交换的序列和几乎一致收敛的性质，鞅不等式是至关重要的。鞅不等式和非对易概率也是分析算子空间OH及其在III型von Neumann代数的准对偶中实现的基本工具。这些技巧类似于Pisier/Shlyaktenko在证明Grothendieck不等式的非对易版本时所使用的技巧。令人惊讶的是，在算子空间的背景下，‘小Grothendieck不等式’的类似只支持一个对数因子。量子力学和海森伯格的测不准原理和数学模型实现了这些现象，不仅改变了我们对世界的看法，也改变了数学学科。经典数学理论的许多非对易(=量子)推广，例如量子群论和非对易概率论。当将经典概念应用到这个非对易框架时，出现了非常有趣的新现象和困难。非对易测度论和von Neumann代数理论提供了大量真正新现象的例子。事实上，冯·诺伊曼研究算子代数(现在称为冯·诺伊曼代数)的动机是为量子力学提供良好的数学基础。在这一传统中，算子空间理论为量化Banach空间提供了正确的语言，这是一个为描述微分方程解的空间而发展起来的概念。例如，使用这种语言，现在可以讨论非对易域的预期退出时间，尽管我们永远看不到这个域的‘点’。从长远来看，这些数学理论提供了新的特征，可以用来理解物理学和其他自然科学中的现象。

AbstractJungeThe starting point of this project on 'Lp estimates in noncommutative probability and analysis' are recent results and techniques from noncommutative martingales inequalities obtained by Pisier/Xu, Randrianantoanina, Junge and Junge/Xu. This new insight enables us to show a noncommutative analog the maximal ergodic theorem for completely positive maps and study square function inequalities in this setting (joint work with LeMerdy/Xu). The formulation and properties are motivated by the theory of Operator Spaces. On the other hand martingale inequalities are crucial in understanding independent, indiscernable and exchangeable sequences in noncommutative Lp spaces and properties of almost uniform convergence. Martingale inequalities and noncommutative probability are also fundamental tools in analyzing the operator space OH and its realization in the predual of type III von Neumann algebras. These techniques are similar to those used by Pisier/Shlyaktenko in proving the noncommutative version of Grothendieck's inequality. Surprising the analog of the 'little Grothendieck inequality' in the context of operator spaces only holds up to a logarithmic factor.Quantum mechanics and Heisenberg's uncertainty principle and mathematical models realizing these phenomena changed not only our perception of the world but also the mathematical discipline. Many noncommutative (=quantum) generalizations of classical mathematical theories for example the theory of quantum groups and noncommutative (=quantum) probability theory. Very interesting new phenomena and difficulties arise when adopting classical concepts to this noncommutative framework. Noncommutative measure theory and the theory of von Neumann algebras provide plenty of examples of genuinely new phenoma. Indeed, von Neumann's motivation for his work on operator algebras (now called von Neumann algebras) was to provide a good mathematical foundation for quantum mechanics. In this tradition the theory of Operator Spaces provides the right language for quantizing Banach spaces, a notion developed to describe the spaces of solutions of differential equations. For example, using this language it is now possible to talk about the expected exit time for a noncommutative domain although we can never see the 'points' of this domain. As a long term perspective these mathematical theories provide new features which may be used to understand phenomena in physics and other natural sciences.