NSF-BSF: Semigroup Operator Algebras - Representations, Boundaries and Dynamics

New Mexico State University, Las Cruces NM

Investigators

Abstract

This project seeks to advance understanding of semigroups within the framework of operator algebras, an area of mathematical analysis. The mathematical field of operator algebras originated in the pioneering work of Murray and von Neumann in the 1930s to develop mathematical foundations for quantum mechanics and has grown into a vital subarea of modern analysis, with connections to many other mathematical fields, including geometry, topology, mathematical physics, and algebra. Operator algebras and the theory of semigroups have enjoyed a fruitful dialogue since the 1990s. Semigroups are algebraic structures in which (like the real numbers) elements can be combined using algebraic operations, but which (unlike the real numbers) generally lack inverse operations. Due to this algebraic structure, semigroups are often useful in modeling “irreversible” phenomena, such as the time evolution of some physical systems, and have proven to be an essential tool across the mathematical sciences. This project will develop new theories and methods to study semigroups and their associated operator algebras through their representations, boundaries and dynamics. The project will also foster international collaboration, enhance the research culture at New Mexico State University, and provide training of graduate and undergraduate students with an emphasis on including students from underrepresented groups. This is a project funded jointly by the National Science Foundation’s Division of Mathematical Sciences, in the Directorate for Mathematical and Physical Sciences (NSF-MPS-DMS), and the Israel Binational Science Foundation (BSF) in accordance with the Memorandum of Understanding between the NSF and the BSF. This project concerns the interrelationships among several classes of semigroup representations and associated operator algebras and dynamics. Unlike groups whose representations are always unitary, semigroups have richer classes of representations as operators on Hilbert spaces. The first goal of this project is to develop a general theory of dilation of isometric covariant representations to characterize boundary representations, and thereby calculate the C*-envelope of universal non-self-adjoint semigroup operator algebras. The focus is on the class of non-Nica-amenable semigroups, whose representations remain highly mysterious. The second goal of this project is to study semigroups from a dynamical perspective, seeking to understand the properties of semigroup operator algebras from the properties of the underlying dynamics. In particular, this project aims to build a general framework to study two-sided semigroup actions, which is motivated by mathematical physics. This project will also investigate the dynamics of self-similar actions, which exhibits new phenomena that generalize many group dynamics. Finally, this project will investigate the representation theory of Artin semigroups, which is a rich class of semigroups with deep connections to various areas of mathematics. In addition to providing concrete examples, the study of Artin semigroups will also contribute to multivariable operator theory and non-commutative geometry. 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.