Approximation Properties of C*-algebras

本项目将讨论C*-代数的近似性质及其与非交换动力学、群论和算子空间的联系。Perron-Frobenius算符是研究密度随时间演化的一种自然而有用的方法。首席研究员将研究经典Perron-Frobenius算子以及非对易Perron-Frobenius算子的动力学，特别关注拓扑熵。在另一个方向上，他将研究完全有界近似性质在约化自由积上的表现，重点是与自由积的类比和群的弱适性。最后，该项目将研究C*-代数的局部线性结构（从算子空间的角度来看）如何控制C*-代数的代数、拓扑和表示理论性质。近似理论是数学中一个非常有用的概念。其基本思想如下：取一个人们试图理解的无限的（因此对人类思维来说难以理解的）数学对象；然后用类似的，但有限的（因此理论上更容易理解的）数学对象来近似这个对象；最后，将所有这些近似合在一起，形成对无限数学对象的清晰理解。在这个建议中的数学对象具有技术名称C*-代数。它们最初是在量子力学的背景下研究的，但后来发现了与几乎所有数学领域的许多联系，例如，与动力系统和群论。这个项目旨在用相同的近似理论工具来处理三个看似独立的数学领域（动力系统、群论和算子空间），所有这些都是在C*代数的背景下进行的。

This project will address approximation properties of C*-algebras and their connection to noncommutative dynamics, group theory, and operator spaces. Perron-Frobenius operators are a natural and useful way to study the time-evolution of densities. The principal investigator will study the dynamics of classical Perron-Frobenius operators as well as noncommutative Perron-Frobenius operators, with a special focus on topological entropy. In another direction, he will investigate how the completely bounded approximation property behaves with respect to reduced free products, focusing on the analogies with free products and weak amenability for groups. Finally, the project will study how the local linear structure (from an operator space point-of-view) of a C*-algebra controls the algebraic, topological, and representation theoretic properties of a C*-algebra.Approximation theory is an extremely useful concept in mathematics. The basic idea is as follows: take an infinite (and therefore difficult for the human mind to comprehend) mathematical object that one seeks to understand; then approximate this object by similar, yet finite (and therefore theoretically easier for the human mind to comprehend) mathematical objects; finally, fit all of these approximations together to compose a clear understanding of the infinite mathematical object. The mathematical objects in this proposal bear the technical name C*-algebras. They were first studied in the context of quantum mechanics but have since realized numerous connections to nearly all areas of mathematics, for example, to dynamical systems and to group theory. This project aims to treat three seemingly separate areas of mathematics (dynamical systems, group theory, and operator spaces) with the same tool of approximation theory, all in the context of C*-algebras.