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

本课题主要研究巴拿赫空间理论及其与算子代数和算子理论的关系。主要考虑的方向是(1)p可积函数/算子的非交换空间，(2)非交换Hardy空间，以及(3)非交换鞅不等式。当p = 1时，可积算子的非交换空间是von Neumann代数的前双元，反映了其基础代数的许多重要性质。在这个研究方向上的基本持久性问题是一个给定的性质是否可以从一个给定的函数空间提升到它的非交换对应物。一个重要的研究方向是研究冯诺依曼代数的前偶的自反子空间的不同结构性质。例如，对于非膨胀映射，这样的空间是否具有不动点属性仍然是未知的。对于Hardy空间，主要目标是发展和扩展从调和分析到非交换设置的工具。在一个稍微不同的方向，正在进行的研究非交换鞅将考虑。主要目标之一是找到不同经典不等式的适当量化，并将这些结果应用于C*代数和冯·诺伊曼代数，以及其他一些相关领域。这项研究代表了跨学科性质的数学分析工作。巴拿赫空间理论是研究无限维向量空间上距离的概念。它为几个数学领域提供了一般框架。函数空间理论对巴拿赫空间理论的发展起了至关重要的作用。本课题研究了一个相对较新的函数空间非交换模拟的概念，其中函数被算子取代。这些空间包括C*代数、冯·诺伊曼代数的前公数等。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也有重要的应用（例如，几何、数学物理和量子力学），所以从许多不同的角度考虑它们是很重要的。非交换概率为量子力学中的概率观点提供了一个可能的框架。

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.