Quantum Probabilistic Methods in Operator Spaces and Applications

算子空间中的量子概率方法及其应用

• 批准号: 0556120
• 负责人: Marius Junge
• 金额: $ 16.2万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0556120&HistoricalAwards=false
• 关键词: Quantum; Probabilistic; Methods; Operator; Spaces

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.

这个关于“非交换概率和分析中的 Lp 估计”的项目的起点是 Pisier/Xu、Randrianantoanina、Junge 和 Junge/Xu 获得的非交换鞅不等式的最新结果和技术。 这一新见解使我们能够展示完全正映射的最大遍历定理的非交换模拟，并研究这种情况下的平方函数不等式（与 LeMerdy/Xu 联合工作）。 公式和性质是由算子空间理论激发的。 另一方面，鞅不等式对于理解非交换 Lp 空间中独立的、不可辨别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 型冯诺依曼代数预置中的实现的基本工具。 这些技术类似于 Pisier/Shlyaktenko 在证明格洛腾迪克不等式的非交换版本时使用的技术。 令人惊讶的是，算子空间中“小格洛腾迪克不等式”的类比仅适用于对数因子。量子力学和海森堡的不确定性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）概括，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个非交换框架时，会出现非常有趣的新现象和困难。非交换测度论和冯诺依曼代数理论提供了大量真正新现象的例子。事实上，冯·诺依曼从事算子代数（现在称为冯·诺依曼代数）工作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，Banach 空间是为描述微分方程解的空间而开发的概念。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，尽管我们永远看不到该域的“点”。从长远来看，这些数学理论提供了新的特征，可用于理解物理学和其他自然科学中的现象。