Interactions between Reversible and Irreversible Operator Algebras

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

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.