Mathematical Sciences: Structure of Vector-Valued Function Spaces and Non-Commutative Function Spaces

拟重点研究两个方向：(1)C*-代数及其相关空间的Banach空间结构；(2)向量值函数空间的研究。在(1)中，他将研究两个领域之间的联系：巴拿赫空间理论和算子代数理论。本方向研究的主要目的是证明Banach空间理论和调和分析的一些经典结果在非交换条件下仍然有效。一个重要的方向是研究与半有限冯诺依曼代数相关的可测算子的对称空间。基本的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保持到它的非交换版本。例如，有限型（分别为有限协型）的（非交换）空间的分类是未知的；存在与给定离散群的riesz子集相关的哪些空间具有Radon-Nikodym性质的问题。另一个方向是研究与有限von Neumann代数的极大有限次对角代数相关的非交换Hardy空间。例如，经典分析中的Szego定理是否对一般的非交换集有效是未知的。如果这些极大子代数的前对偶是有限共型的，它仍然是开的。对于非交换Hardy空间和Popescu引入的非交换盘代数，也考虑了同样的研究。我们对非交换C*-代数的巴拿赫空间结构的认识（尽管最近有一些深入的研究）仍然不令人满意；提议者还将处理一些从经典案例中自然产生的问题。例如，C*-代数上不同类别的有界算子的行为还远远没有被很好地理解。几个中间结果，如绝对和算子的分解和绝对和算子的紧性，已经得到，暗示一般的非交换C*-代数应该表现（在许多方面）像它们的交换对应物。这部分的主要动机是与这些代数相关的某些问题在用巴拿赫空间语言表述时变得更加透明。第(2)部分可以看作是测度理论与巴拿赫空间理论之间的紧密联系。关于Bochner空间所保留的恒久性，仍有许多问题有待解决。本文将集中讨论包含经典函数空间的空间的性质、前奏曲的唯一性和强正则性。本计画的目的是提高对不同分析领域之间的联系的理解：巴拿赫空间理论、调和分析和算子代数理论。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，数学物理、几何、量子力学），所以从许多不同的角度看待它们是很重要的。

9703789 Randrianantoanina The proposer intends to focus on two lines of research: (1) Banach space structures of C*-algebras and related spaces; (2) studies of vector-valued function spaces. In (1), he will study some connections between the two fields: Banach space theory and operator algebra theory. The primary goal of this direction of research is to show that some classical results from Banach space theory and harmonic analysis are still valid for the non-commutative setting. One important direction is the investigation of symmetric spaces of measurable operators associated with semi finite von Neumann algebras. The basic permanence question is whether or not a given property can be preserved from a given function space to its non commutative version. For instance, the classification of (non commutative) spaces of finite type (respectively of finite cotype) is not known; there is the question of which spaces have the Radon-Nikodym properties associated with Riesz-subsets of a given discrete group. Another direction is the study of non commutative Hardy spaces associated with maximal finite subdiagonal algebras of finite von Neumann algebras. For example, it is unknown if Szego's theorem from classical analysis is valid for the general non commutative setting. It is still open if the predual of such maximal subalgebras are of finite cotype. The same kind of investigation will also be considered for non commutative Hardy spaces and the non commutative disc algebras introduced by Popescu. Our knowledge of Banach space structures of non commutative C*-algebras (despite some recent intensive studies) is still less than satisfactory; the proposer also will tackle some of the questions that arise naturally from the classical case. For instance, the behavior of different classes of bounded operators on C*-algebras are far from being well understood. Several intermediate results, such as factorizations of absolutely summing operators and compactness of absolutely summing operators, were already obtained, hinting that in general non commutative C*-algebras should behave (in many ways) like their commutative counterparts. The main motivation for this part is that certain problems related to these algebras become more transparent when formulated in Banach space language. Part (2) can be viewed as close connections between measure theory and Banach space theory. Many questions remain unresolved on permanence properties preserved by Bochner spaces. The proposer will concentrate on properties of spaces containing classical function spaces, uniqueness of preduals, and strong regularity. The purpose of this project is to improve the understanding of the connections between different field of analysis: Banach space theory, harmonic analysis, and operator algebra theory. 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, mathematical physics, geometry, quantum mechanics), so it is important to view them from many different angles.