Interactions between Reversible and Irreversible Operator Algebras

算符代数最初是泛函分析的一个分支，现已发展成为独立的领域，并影响到数学和物理的各个领域，包括动力系统、群论、几何、量子力学和量子信息论。这项研究是由冯-诺伊曼在30年代发起的，受到与量子力学相关的数学的启发，起源于单算符论和复变分析。如今，算符代数被成功地用来为经典数学理论提供新的视角，并为物理理论奠定基础。这个项目的主要目标是在“不可逆”和“可逆”算子代数这两个看似不同的理论之间建立和扩展现有的相互作用。这些相互作用导致了这一领域的实质性突破，对上述数学和物理领域产生了影响。更具体地说，该项目旨在将非自伴算子代数、Arveson非交换边界理论和C*-代数的技术结合在一起，涉及到半群和群论、膨胀理论、非交换动力系统以及C*-代数的表示的分类，直到酉等价。该项目有三个主要目标。第一个目标是在Kalantar和Kennedy工作的启发下，利用Hamana的内射包络理论来刻画半群的边界商C*-代数的核性、单性等C*-代数性质。第二个目标是将Exel的思想引入到分次和余作用C*-代数上，将Katsoulis的PI关于乘积系统中C*-包络的刻画的工作推广到非阿贝尔的情况。第三个目标是继续Davidson，B.Li和PI开始的研究，其中非自伴算子代数被用来区分与有向图相关的C*-代数的表示。上述目标的进展将加强可逆和不可逆理论之间的双向互动，并将导致解决该领域几个公开问题的新方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Originally a branch of functional analysis, the field of operator algebras has grown to stand on its own and influence diverse areas in mathematics and physics including dynamical systems, group theory, geometry, quantum mechanics and quantum information theory. The study was initiated by von-Neumann in the 30s, motivated by mathematics related to quantum mechanics, and originates from single operator theory and complex analysis. These days, operator algebras are successfully used to provide novel perspectives for classical mathematical theories and to lay the foundations for physical theories. The main overarching goal of this project is to create and expand upon existing interaction between the seemingly separate theories of "irreversible" and "reversible" operator algebras. Such interactions have led to substantial breakthroughs in the field, with impact to the aforementioned areas of mathematics and physics.More specifically, the project aims to bring together techniques from non-self-adjoint operator algebras, Arveson's non-commutative boundary theory and C*-algebras to bear on semigroup and group theories, dilation theory, non-commutative dynamical systems and classification of representations of C*-algebras up to unitary equivalence. The project has three main goals. The first goal, inspired by work of Kalantar and Kennedy, is to find characterizations of nuclearity, simplicity and other C*-algebraic properties of boundary quotient C*-algebras of semigroups using Hamana's injective envelope theory. The second goal is to import ideas of Exel on graded and coaction C*-algebras to extend work of the PI with Katsoulis on characterizations of the C*-envelope in the context of product systems to the non-abelian case. The third goal is to continue the investigation initiated by Davidson, B. Li and the PI where non-self-adjoint operator algebras are used to distinguish representations of C*-algebras associated to directed graphs. Progress on the above-mentioned goals will enhance the two-way interaction between the reversible and irreversible theories and will lead to new approaches for resolving several open problems in the field.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.