Geometric Structure of Operator Spaces

目前的项目旨在研究算子空间的几何性质，重点研究两个主题：具有规定性质的算子空间的存在性，以及费米子代数的结构及其相关空间。构造具有给定性质的空间的问题最早是由格罗滕迪克提出的，后来引起了许多有成就的数学家的注意，如高尔斯和毛里。然而，这一领域的许多问题仍未解决，其中最主要的是所谓的“方-立方问题”：一个空间的存在性与其平方不同构，但其平方与立方同构。我们计划使用算子空间的“构建块”来解决这类问题。在研究费米子代数时，我们计划发现它与函数空间共有的性质。特别地，我们将研究这个代数的补和完全补子空间。在过去的一个世纪里，物理学的关键进步之一是量子力学的创造和发展。量子力学的主要数学工具是用希尔伯特空间（它们可以被认为是无限矩阵）上的算子代替标量（数字）。最初，数学家和物理学家只研究单个算子，但后来需要考虑整个算子集变得明显。冯·诺伊曼、盖尔芬德和其他人的研究导致了C*代数理论的发展。最近，算子空间（也称为“非交换”或“量子化巴拿赫空间”）从对C*-代数上的映射的研究中产生。结果表明，算子空间为研究算子代数提供了一个合适的框架。事实上，算子空间技术已经解决了算子理论中几个长期存在的问题。目前的项目涉及两个主题：具有规定属性的算子空间的存在性；以及对经典空间的算子空间结构的研究，例如费米子代数（与质子或中子等亚原子粒子的行为有关）。如果成功，这项研究将促进我们对算子空间的理解，并有可能增强我们对物理现象的认识。

The current project aims to investigate geometric properties of operator spaces, with the emphasis on two topics: the existence of operator spaces with prescribed properties, and the structure of the Fermion algebra and spaces related to it. The problem of constructing spaces with given properties was first posed by Grothendieck, and later attracted attention of many accomplished mathematicians, such as Gowers and Maurey. However, many questions in this area remain open, chief among them the so-called ``square-cube problem'': the existence of a space which is not isomorphic to its square, but such that its square is isomorphic to its cube. We plan to approach this class of problems using operator space ``building blocks''. In investigating the Fermion algebra, we plan to discover what properties it shares with spaces of functions. In particular, we will investigate the complemented and completely complemented subspaces of this algebra.One of the key advances in physics over the past century was the creation and development of quantum mechanics. The main mathematical tool of quantum mechanics is replacing scalars (numbers) by operators on a Hilbert space (they can be thought of as infinite matrices). Initially, the mathematicians and physicists investigated only single operators, but later the need to consider whole sets of operators became apparent. Research by von Neumann, Gelfand, and others led to the development of the theory of C*-algebras. More recently, operator spaces (also called ``non-commutative'' or ``quantized Banach spaces'') arose from the study of maps on C*-algebras. It turns out that operator spaces provide an appropriate framework for studying algebras of operators. In fact, several long-standing problems in operator theory have been solved using operator space techniques. The current project deals with two topics: the existence of operator spaces with prescribed properties; and an investigation of the operator space structure of classical spaces, such as the Fermion algebra (related to the behavior of sub-atomic particles such as protons or neutrons). If successful, this research will advance our understanding of operator spaces, and potentially, enhance our knowledge of physical phenomena.