Classifying *-Homomorphisms

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

This project relates to the classification and structure of amenable operator algebras. Operator algebras is an area of mathematics which began with the work of Murray and von Neumann in the 1930's motivated in part by Heisenberg's approach to quantum mechanics in terms of infinite matrices. Von Neumann's development of operators on a Hilbert space put Heisenberg's ideas on rigorous foundations. In this theory, the observable data is represented by certain operators (i.e. infinite matrices) on a Hilbert space. One of the most famous concepts of quantum mechanics is Heisenberg's uncertainty principle that the speed and position of a particle cannot be known simultaneously. The mathematically rigorous version of this statement is that the operators P and Q which measure position and momentum do not commute; i.e. PQ and QP are not equal (there is, however, a precise formula relating P and Q). Operator algebras is the study of algebraic relations between collections of operators. Amenable operators algebras form a particularly important class of operators algebras. In recent years, there has been a substantial work in classifying simple amenable operator algebras. In a certain sense, the simple examples are the basic building blocks of the theory. The goal of this project is to understand the relations between the simple amenable operator algebras (such as ways one can be embedded into the other) and the symmetries of such algebras and to exploit these ideas to uncover structural information about operator algebras as a whole. This project also will be contributing to the education of the US workforce through the training of graduate students. More technically, recent progress in Elliott's Program shows separable, simple, nuclear, regular C*-algebras in the UCT class are classified up to isomorphism via their operator K-theory groups, their trace simplex, and the pairing between them. This should be viewed as a direct analogue to the Connes-Haagerup classification of separably acting injective factors in terms of their type and flow of weights. The Connes-Haagerup classification has become a corner stone of modern von Neumann algebra theory, and the C*-algebraic analogue may be expected to have an equally important role in C*-algebra theory. In recent joint work with Carrion, Gabe, Tikuisis, and White, the PI showed that embeddings of "classifiable" C*-algebras are determined up to approximate unitary equivalence by K-theoretic data. The aim of this project is to refine and expand on these techniques with an eye towards equivariant classification and non-simple classification, as well as applications to the regularity theory, such as computing the nuclear dimension of a new classes of C*-algebras. 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.