Operator Algebras and Topological Invariants

算子代数和拓扑不变量

• 批准号: 1101305
• 负责人: Marius Dadarlat
• 金额: $ 31.96万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101305&HistoricalAwards=false
• 关键词: Operator; Algebras; Topological; Invariants

This project is guided by the premise that the ideas and the methods of algebraic topology, understood in a broad sense, will continue to play a key role in the study of operator algebras. The underlying theme of the research is to devise new techniques and tools in the deformation theory of operator algebras. The idea of deformations of algebras is intimately related to K-theory, as is revealed by the development of E-theory. In particular one can read the K-homology of a locally compact connected space X from matricial deformations of the algebra of continuous functions on X. In what may be regarded as a counterpart of this property in condensed-matter physics, Kitaev has proposed very recently a classification of topological insulators that is based on real K-homology and that relies ultimately on a classification up to approximate unitary equivalence of matricial deformations of spaces. A first objective of the project is to study invariants of C*-algebras that integrate the primitive spectrum along with homological invariants. This is naturally tied to the idea of deformation and the theory of continuous fields and will involve the study of KK-theory and E-theory parametrized by spaces that are not necessarily Hausdorff. A second objective is to investigate the existence of deformations of group C*-algebras into matrix algebras with the property that they detect the K-theory of the group C*-algebras. In addition, the principal investigator aims to provide formulas for computing invariants associated with these deformations. A third objective is to study invariants of simple C*-algebras that capture properties not detected by K-theory or tracial states. In particular, the project will investigate rigidity properties of simple C*-algebras.The physics of elementary particles has led to new mathematical theories in which numerical functions are replaced with infinite arrays of numbers or matrices. Matrices can be multiplied, but unlike numerical multiplication, the order of the factors is essential, so that A times B is not always equal to B times A. This property of matrix multiplication, called noncommutativity, is crucial for the interpretation of very complex phenomena in quantum mechanics that might initially appear completely counter-intuitive (e.g., Heisenberg's uncertainty principle). The passage from numbers to matrices often involves a process of deformation in which spatial structures are discretized in such a way that the possible twists of surfaces (or even more complicated objects) are captured in the form of numerical invariants. This project is concerned with the theory of such deformations and connects classical geometry with noncommutative geometry, a theory that proposes a reconstruction of the general concept of space that is better aligned with the principles of quantum physics than is traditional geometry. Moreover, the deformation theory that the principal investigator will study provides mathematical tools relevant to physical phenomena that are observed in certain special materials. These materials behave as insulators in their interiors, while permitting the movement of charges on their surfaces, due to exotic metallic states localized there. Such remarkable new materials, known as "topological insulators," have already had a considerable impact on condensed matter physics and are expected to find technological applications in magneto-electronics, an emerging field in which spin currents are used instead of charge currents, and in quantum computing.

该项目的指导是以这样的前提为指导，即代数拓扑的思想和方法在广义上将继续在操作员代数研究中继续发挥关键作用。该研究的基本主题是在操作员代数的变形理论中设计新技术和工具。代数变形的想法与K理论密切相关，正如E理论的发展所揭示的那样。 In particular one can read the K-homology of a locally compact connected space X from matricial deformations of the algebra of continuous functions on X. In what may be regarded as a counterpart of this property in condensed-matter physics, Kitaev has proposed very recently a classification of topological insulators that is based on real K-homology and that relies ultimately on a classification up to approximate unitary equivalence of matricial deformations of空间。该项目的第一个目的是研究C* - 代数的不变性，这些C* - 代数将原始频谱与同源不变式一起整合。这自然与变形的概念和连续场的理论有关，并且将涉及对不一定是Hausdorff的空间参数的KK理论和E理论的研究。第二个目标是研究C* - 代数组的变形在矩阵代数中，并具有检测到C*-Algebras组K理论的属性。此外，主要研究人员旨在为计算与这些变形相关的计算不变性提供公式。第三个目标是研究简单的C* - 代数的不变性，该代数捕获了K理论或奇异状态未检测到的特性。特别是，该项目将研究简单的C*-Algebras的刚性特性。基本粒子的物理学已导致新的数学理论，其中数值函数被无限数字或矩阵替换。 Matrices can be multiplied, but unlike numerical multiplication, the order of the factors is essential, so that A times B is not always equal to B times A. This property of matrix multiplication, called noncommutativity, is crucial for the interpretation of very complex phenomena in quantum mechanics that might initially appear completely counter-intuitive (e.g., Heisenberg's uncertainty principle).从数字到矩阵的段落通常涉及一个变形过程，在该过程中，空间结构被离散化的方式使可能的表面曲折（甚至更复杂的对象）以数值不变性的形式捕获。该项目与这种变形的理论有关，并将经典的几何形状与非共同的几何形状联系起来，该理论提出了对一般空间概念的重建，该理论与传统几何学的原理更好地与量子物理原理保持一致。此外，主要研究者将研究的变形理论提供了与某些特殊材料中观察到的物理现象相关的数学工具。这些材料在其内部的室内表现为绝缘子，同时允许其表面上的指控移动，这是由于外来的金属状态所在。这种非凡的新材料（称为“拓扑绝缘子）”已经对凝聚的物理学产生了相当大的影响，并有望在磁电子学中找到技术应用，这是一个新兴领域，其中使用了自旋电流代替电荷电流，以及在量子计算中。