NSF-BSF: C*-algebras and Dynamics Beyond the Elliott Program

NSF-BSF：艾略特纲领之外的 C* 代数和动力学

• 批准号: 2400332
• 负责人: Norman Phillips
• 金额: $ 34.33万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-08-01 至 2027-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2400332&HistoricalAwards=false
• 关键词: NSF; BSF; algebras; Dynamics; Beyond

A C*-algebra is a kind of mathematical object which, for example, appears in quantum mechanics. Simple C*-algebras are those that cannot be broken apart into smaller ("simpler") C*-algebras. The largest part of this project is about when a simple C*-algebra is isomorphic to its opposite algebra, that is, mathematically the same as what might be thought of as its mirror image. For an example from everyday life, an ordinary sock is the same as its mirror image, since a sock which fits on a right foot will fit equally well on the left foot. A glove isn't like that: whatever one does, a right glove will not fit on a left hand. A nonsimple C*-algebra can be made of very elementary parts, but put together in a tricky way, so as to not be isomorphic to its opposite. Simple C*-algebras which are not separable or not nuclear ("too large", but in different senses) can also fail to be isomorphic to their opposites. On the other hand, simple C*-algebras covered by the Elliott classification program are isomorphic to their opposites. A long-term goal of the project is to exhibit a simple separable nuclear C*-algebra which is not isomorphic to its opposite. Such an algebra could not be covered even by any proposed expansion of the Elliott program. The project will also contribute to US workforce development through the training of graduate and undergraduate students.The intended example is a simple unital AH algebra with fast dimension growth. The intended proof that it is not isomorphic to its opposite depends on nonexistence theorems for certain homomorphisms from one matrix algebra over the algebra of continuous functions on a compact space to a different matrix algebra over the continuous functions on a different compact space. When the second matrix size is large enough, all homomorphisms not ruled out for fairly obvious reasons actually exist. When it is small, known obstructions rule out most homomorphisms. The application requires information about an intermediate range. Here, even the simplest case, asked by Blackadar over 30 years ago, remains open; understanding this case is a necessary preliminary step. This case can almost certainly be settled by computations in rational homotopy theory, a new use of algebraic topology in C*-algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

C* 代数是一种数学对象，例如出现在量子力学中。简单 C* 代数是那些不能分解为更小的（“更简单”）C* 代数的代数。该项目的最大部分是关于一个简单的 C* 代数何时与其相反代数同构，即在数学上与其镜像相同。举个日常生活中的例子，一只普通的袜子与其镜像是一样的，因为适合右脚的袜子同样适合左脚。手套则不然：无论做什么，右手手套都不适合左手。非简单的 C* 代数可以由非常基本的部分组成，但以一种棘手的方式组合在一起，以免与其相反的同构。不可分或非核的简单 C* 代数（“太大”，但在不同的意义上）也可能无法与其相反的同构。另一方面，艾略特分类程序涵盖的简单 C* 代数与其相反的代数是同构的。该项目的长期目标是展示一个与其相反的简单可分离核 C* 代数。即使艾略特纲领的任何拟议扩展也无法涵盖这样的代数。该项目还将通过培训研究生和本科生为美国劳动力发展做出贡献。预期的示例是一个具有快速维度增长的简单单位 AH 代数。它与其对立面不同构的预期证明取决于从紧致空间上的连续函数代数上的一个矩阵代数到不同紧致空间上的连续函数上的不同矩阵代数的某些同态的不存在定理。当第二矩阵大小足够大时，由于相当明显的原因而未被排除的所有同态实际上都存在。当它很小时，已知的障碍排除了大多数同态。该应用程序需要有关中间范围的信息。在这里，即使是 Blackadar 在 30 多年前提出的最简单的案例，也仍然悬而未决。了解此案是必要的初步步骤。这种情况几乎可以肯定可以通过有理同伦理论的计算来解决，这是代数拓扑在 C* 代数中的新用途。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。