Structural Invariants for Higher-Rank Graphs

University Of Montana, Missoula MT

Investigators

Abstract

The theory of C*-algebras was invented in the 1930s and 1940s as a mathematical model for quantum mechanics. In addition to their connections with physics, C*-algebras provide a useful framework for studying a variety of mathematical objects. Applications of C*-algebras beyond physics are also beginning to arise; for example, one can use C*-algebras to study both quantum information theory, and directed graphs, or networks. Conversely, the C*-algebras associated to directed graphs have proved to be key examples which have enhanced our understanding of the class of C*-algebras as a whole. This project focuses on a generalization of directed graphs -- higher-rank graphs -- and their associated C*-algebras. A better understanding of the C*-algebras associated to higher-rank graphs will strengthen our understanding of C*-algebras as a whole, and also their applicability to other areas of mathematics. In addition to involving graduate and undergraduate students in research about higher-rank graph C*-algebras, this project will also enhance the intellectual opportunities available to mathematicians at the University of Montana and at nearby universities, in two ways. First, it initiates an exchange program for research-level graduate students at the University of Montana, to enable them to spend a semester studying at a university which is offering an advanced graduate course or seminar that complements the student's research interests. Second, the principal investigator will host national and international research scholars at the University of Montana, and will facilitate these scholars' visits to other regional institutions during their stay in Montana. Higher-rank graphs were introduced by Kumjian and Pask in 2000, in the hopes that their associated C*-algebras, like graph C*-algebras, would provide important insights about C*-algebras more generally. However, the structure of higher-rank graph C*-algebras is much more intricate than that of graph C*-algebras, and this complexity has limited the applicability of higher-rank graph C*-algebras to other areas of mathematics up to now. To reverse this trend, this project will (1) improve our ability to recognize higher-rank graph C*-algebras in other contexts; and (2) develop new, stronger tools for analyzing higher-rank graph C*-algebras. Towards the first goal, the principal investigator will clarify the relationship between higher-rank graph C*-algebras and their twisted counterparts, as well as investigate connections between higher-rank graph C*-algebras and other branches of mathematics (such as Lie algebras and lattices, and wavelets and multiresolution analyses). The second goal will be achieved by studying invariants of higher-rank graph C*-algebras such as their K-theory, groupoid structure, KMS states, and Cartan subalgebras. 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.