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

C*代数是一种数学对象，例如出现在量子力学中。简单C*代数是那些不能被分解成更小的（“更简单的”）C*代数的代数。这个项目最大的部分是关于当一个简单的C*代数与它的相反代数同构时，也就是说，在数学上与它的镜像是相同的。举一个日常生活中的例子，一只普通的袜子和它的镜像是一样的，因为一只适合右脚的袜子也同样适合左脚。手套不是这样的：无论你做什么，右手手套都不适合左手。一个非简单的C*代数可以由非常基本的部分组成，但以一种巧妙的方式组合在一起，这样就不会与它的对立面同构。不可分离或非核（“太大”，但在不同的意义上）的简单C*-代数也不能与它们的对立面同构。另一方面，Elliott分类程序所涵盖的简单C*-代数与它们的对立面是同构的。该项目的长期目标是展示一个简单的可分离核C*-代数，它与它的对立面不同构。这样的代数甚至不能被任何提出的艾略特计划的扩展所涵盖。该项目还将通过培训研究生和本科生为美国的劳动力发展做出贡献。预期的示例是一个具有快速维数增长的简单一元AH代数。要证明它与它的对立面不同构，取决于紧空间上连续函数代数上的一个矩阵代数到紧空间上连续函数上的另一个矩阵代数的同态的不存在性定理。当第二个矩阵的大小足够大时，由于相当明显的原因而没有被排除的所有同态实际上都存在。当它很小时，已知的障碍物排除了大多数同态。应用程序需要有关中间范围的信息。在这里，即使是30多年前Blackadar提出的最简单的案例，也仍然悬而未决；了解这种情况是必要的初步步骤。这种情况几乎可以肯定地通过有理同伦理论的计算来解决，这是代数拓扑在C*-代数中的一种新应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.