Geometry of Operator Spaces

Oikhberg计划继续他在算子空间几何的几个方面的工作。其中一项工作是处理可分离的扩展属性。这里最有趣的问题是K(可分Hilbert空间上的紧算子空间)是否在包含它的每个可分算子空间中是可补的(这是Sobczyk的一个经典结果的“非对易”版本)。一个相关的问题是给出了在每个可分局部自反算子超空间(对应于Zippin的收敛序列空间刻画的算子空间)中完全可补的可分局部自反算子空间的完整刻画。另一个研究方向是研究极大算子空间(交换C*-代数的对偶的商)。PI计划确定极大空间的n维子空间集的基数，以及2n维交换C*-代数的对偶的n维子空间的结构(本着Kashin的工作精神)。这项研究的可能应用包括更好地理解函数空间的结构，希尔伯特空间上的算符空间，以及一般的Banach空间几何(局部自反性的扩展)。量子力学的关键数学工具之一是用希尔伯特空间上的算符(它们可以被视为无限矩阵)来代替标量(数字)。在这个方向上的进一步探索导致了C*-代数理论的发展，后来又引入了算子空间(也称为“非交换的”或“量化的Banach空间”)的概念。通过将算子空间嵌入到Hilbert空间H上的有界线性算子的C*-代数B(H)中，人们可以将算子空间看作是具有附加结构的Banach空间。在过去的十年里，算子空间的研究结合了Banach空间理论和C*-代数理论的思想和技巧，取得了非常迅速的进展。它已经为算子理论的一些长期存在的问题提供了答案。所提出的研究涉及两个主题：经典Banach空间结果的非对易类比的存在性；以及Banach空间方法在算子空间中的应用。如果成功，这项研究将促进我们对Banach空间和算子空间之间的联系的理解，并潜在地增强我们对物理现象的知识。

AbstractOikhbergThe PI plans to continue his work on several aspects of geometry of operator spaces. One line of work deals with separable extension properties. Here the most intriguing problem is whether K (the space of compact operators on a separable Hilbert space) is complemented in every separable operator space containing it (this is a "non-commutative" version of a classical result of Sobczyk). A related issue is giving a complete description of separable locally reflexive operator spaces which are completely complemented in every separable locally reflexive operator superspace (an operator space counterpart of Zippin's characterization of the space of convergent sequences). Another direction of research is the study of maximal operator spaces (quotients of duals of commutative C*-algebras). The PI plans to determine the cardinality of the set of n-dimensional subspaces of maximal spaces, as well as the structure of n-dimensional subspaces of the dual of a 2n-dimensional commutative C*-algebra (in the spirit of Kashin's work). Possible applications of the proposed research include better understanding of the structure of function spaces, of the space of operators on a Hilbert space, and Banach space geometry in general (extensions of local reflexivity).One of the key mathematical tools of quantum mechanics is replacing scalars (numbers) by operators on a Hilbert space (they can be thought of as infinite matrices). Further exploration in this direction led to the development of the theory of C*-algebras, and, later, to the introduction of the notion of operator space (also called "non-commutative" or "quantized Banach space"). One can view an operator space as a Banach space with additional structure, induced by its embedding into the C*-algebra B(H) of bounded linear operators on a Hilbert space H. The investigation of operator spaces has advanced very rapidly over the last ten years, combining ideas and techniques from Banach space theory and the theory of C*-algebras. It has already produced answers to some long-standing problems of Operator Theory. The proposed research deals with two themes: the existence of non-commutative analogs of classical Banach space results; and the use of Banach space methods in the operator space case. If successful, this research will advance our understanding of connections between Banach and operator spaces, and potentially, enhance our knowledge of physical phenomena.