Banach Space Structures of Non-commutative L^p-spaces and Non-commutative Martingale Inequalities

这一研究项目将集中在Banach空间理论及其与算子代数和算子理论的联系上。要考虑的主要方向是(1)p-可积函数/算子的非交换空间的方面，(2)非交换Hardy空间，和(3)非交换鞅不等式。如果p等于1，则可积算子的非交换空间是von Neumann代数的预对偶，它反映了基础代数的许多重要性质。这个研究方向的基本持久性问题是，给定的性质是否可以从给定的函数空间提升到它的非对易对应的函数空间。研究von Neumann代数的自反子空间的不同结构性质是需要考虑的一个重要方向。例如，这类空间是否具有非扩张映射的不动点性质仍是未知的。对于Hardy空间的情况，主要目标是开发和扩展工具，从调和分析到非对易环境。在一个稍有不同的方向上，将考虑正在进行的关于非交换鞅的研究。主要目标之一是找到不同经典不等式的适当量子化，并将这些结果应用到C*-代数和von Neumann代数以及其他一些相关领域。这项研究代表了数学分析的跨学科性质的工作。Banach空间理论是本提案的主要主题，研究无限维向量空间上的距离概念。它为数学的几个领域提供了总体框架。函数空间理论对几十年来Banach空间理论的发展起到了至关重要的作用。本课题研究了一种相对较新的函数空间的非对易类比概念，即用算子代替函数。这些空间包括C*-代数、von Neumann代数的前导等。C*-代数是数学中最重要的结构之一。它们在科学的其他部分(例如，几何学、数学物理和量子力学)都有重要的应用，所以从许多不同的角度来考虑它们是很重要的。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。

This research project will focus on problems dealing with Banach space theory and its connections with operator algebras and operator theory. The main directions to be considered are (1) aspects of non-commutative spaces of p-integrable functions/operators, (2) non-commutative Hardy spaces, and (3) non-commutative martingale inequalities. If p is equal to 1, the non-commutative space of integrable operators is the predual of von Neumann algebra and reflects many important properties of the underlying algebras. The basic permanence question in this direction of research is whether or not a given property can be lifted from a given function space to its non-commutative counterpart. An important direction to be considered is the study of different structural properties of reflexive subspaces of preduals of von Neumann algebras. For instance, it is still unknown if such spaces have the fixed point property for non expansive maps. For the case of Hardy spaces, the primary goal is to develop and extend tools from harmonic analysis to the non-commutative settings. In a somewhat different direction the ongoing research on non-commutative martingales will be considered. One of the main goals is to find the appropriate quantization of different classical inequalities, and apply these results to C*-algebras and von Neumann algebras, as well as to some other related areas.This research represents work of an interdisciplinary nature on mathematical analysis.Banach space theory, which is the main topic of this proposal, studies notions of distances on infinite dimensional vector spaces. It provides general framework for several fields of mathematics. The theory of function spaces played a crucial role in the development of Banach space theory for several decades. The current project studies a relatively new concept of non-commutative analogue of function spaces in which functions are replaced by operators. These spaces includes C*-algebras, preduals of von Neumann algebras among many others. C*-algebras turn out to be one of the most important structures in mathematics. They have significant applications to other parts of sciences (for examples, geometry, mathematical physics and quantum mechanics),so it is important to consider them from many different point of view. Non-commutative probability provides one possible framework for a probabilistic viewpoint in quantum mechanics.