The PI will investigate the structure of Hilbert modules over C*-algebras, and its bearing on the structure of the algebras themselves. Hilbert modules have recently been shown to carry vastly more information than finitely generated projective modules over C*-algebras, even for simple algebras. This project aims to use this fact in studying several aspects of C*-algebra theory: the classification of C*-dynamical systems by their K-theory, the reconciliation of classical and dynamical covering dimension via a Hilbert module invariant of the associated C*-algebras, and the pursuit of a tensorial absorption theorem for nuclear simple separable C*-algebras which will provide a generalization of Kirchberg?s absorption theorem for purely infinite simple C*-algebras. The over-arching theme of the proposal is at the interface of dynamics (systems equipped with a time evolution) and the very natural desire to classify mathematical objects. We aim to use tools from functional analysis to reconcile and interpolate between the properties of classical spaces (no time evolution), periodic spaces (time evolution in which each point follows a finite cycle), and fully dynamic spaces (arbitrary time evolution). Conversely, we will study how the properties of dynamic spaces manifest themselves after translating these spaces into families of operators (infinite-dimensional matrices, if you like), and will work to determine when two such families are essentially the same.
The aim of this project was to investigate the way in which collections of operators--an operator is an object that transforms the space around us in a predictable way--are governed by affiliated objects known as Hilbert modules. The project was ultimately quite successful, as it was partly responsible for new and powerful classification results for collections of operators. Some of these collections arise from quite natural objects, such as a turbulent time evolution on a 3-dimensional sphere. From the intellectual merit standpoint, our project was successful: 10 peer-reviewed articles, some in very highly rated journals, were produced over the course of the project. The results of our project were disseminated at several international conferences, some of which had our project's goals as a major theme and some of which were organized by the PI. The PI began the project as the supervisor of 4 Ph.D. students, one of whom has since graduated. A postdoctoral fellow was also supervised; he now has a tenure-track position. Perhaps the most significant broader impacts under this project came from the PI's ability to successfully apply for conferences at 3 international research institutes. These institutes covered the local accommodation costs of young US researchers, while the PI used funds from this project to support their travel.
