Quantum Probabilistic Methods in Operator Spaces and Applications

本课题“非交换概率和分析中的Lp估计”的出发点是由Pisier/Xu、Randrianantoanina、Junge和Junge/Xu等人从非交换鞅不等式中获得的最新结果和技术。这一新发现使我们能够证明完全正映射的极大遍历定理的非交换类比，并在这种情况下研究平方函数不等式（与LeMerdy/Xu合作）。该公式和性质是由算子空间理论推动的。另一方面，鞅不等式对于理解非交换Lp空间中独立的、不可分割的和可交换的序列以及几乎一致收敛的性质是至关重要的。鞅不等式和非交换概率也是分析算子空间OH及其在III型von Neumann代数前序中的实现的基本工具。这些技术类似于Pisier/Shlyaktenko在证明Grothendieck不等式的非交换版本时所使用的技术。令人惊讶的是，在算子空间中，“小格罗滕迪克不等式”的类比只适用于对数因子。量子力学和海森堡的测不准原理以及实现这些现象的数学模型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交换（=量子）推广，例如量子群理论和非交换（=量子）概率论。当将经典概念应用于这种非交换框架时，会出现非常有趣的新现象和困难。非交换测度理论和冯·诺伊曼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一个良好的数学基础。在这个传统中，算子空间理论为量化巴拿赫空间提供了合适的语言，巴拿赫空间是用来描述微分方程解的空间的一个概念。例如，使用这种语言，现在可以讨论非交换域的预期退出时间，尽管我们永远无法看到该域的“点”。从长远的角度来看，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的现象。

The 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.