Interactions between Reversible and Irreversible Operator Algebras

最初是泛函分析的分支，算子代数领域已经发展成为独立的，并影响了数学和物理学的各个领域，包括动力系统，群论，几何，量子力学和量子信息理论。这项研究是由冯·诺依曼在30年代发起的，受到与量子力学相关的数学的启发，起源于单算符理论和复分析。这些天来，算子代数被成功地用于经典数学理论提供新的视角，并奠定了物理理论的基础。这个项目的主要总体目标是创建和扩展“不可逆”和“可逆”算子代数的看似独立的理论之间的现有相互作用。更具体地说，该项目旨在将非自伴算子代数、Arveson的非交换边界理论和C*-代数的技术结合起来，以影响半群和群论、膨胀理论、非交换动力系统和C*-代数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，受Kalantar和Kennedy工作的启发，利用Hamana的内射包络理论，找到半群的边界商C*-代数的核性、单性和其他C*-代数性质的刻画。第二个目标是导入Exel关于分次和互作用C*-代数的思想，以扩展PI与Katsoulis关于在乘积系统的背景下C*-包络的特征的工作到非阿贝尔情况。第三个目标是继续戴维森，B发起的调查。Li和PI，其中非自伴算子代数用于区分与有向图相关联的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的几个开放问题的新方法。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.