Quantum Probabilistic Methods in Operator Spaces and Applications

这个项目的出发点“Lp估计在非交换概率和分析”是最近的结果和技术从非交换鞅不等式获得的Pisier/徐，Randrianantoanina，Junge和Junge/徐。 这一新的见解使我们能够显示一个非交换模拟的最大遍历定理完全正的地图和研究平方函数不等式在这种设置（联合工作与LeMerdy/徐）。 公式和性质的动机是由算子空间理论。 另一方面，鞅不等式对于理解非交换Lp空间中独立的、不可分辨的和可交换的序列以及几乎一致收敛的性质至关重要。鞅不等式和非交换概率也是分析算子空间OH及其在III型冯诺依曼代数的预对偶中实现的基本工具。 这些技术类似于皮西尔/什利亚坚科在证明格罗滕迪克不等式的非交换版本中所使用的技术。 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对易（=量子）推广，例如量子群理论和非对易（=量子）概率论。当采用经典概念到这个非对易框架时，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，冯诺依曼的动机，他的工作算子代数（现在称为冯诺依曼代数）是提供一个良好的数学基础，量子力学。 在这个传统中，算子空间理论为量化Banach空间提供了正确的语言，这是一个用来描述微分方程解的空间的概念。例如，使用这种语言，现在可以讨论一个非交换整环的期望退出时间，尽管我们永远看不到这个整环的“点”。作为一个长期的观点，这些数学理论提供了新的功能，可用于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.