Hilbert modules and the structure of C*-algebras

Purdue University, West Lafayette IN

Investigators

Abstract

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.