Operator Algebras before and after Jiang-Su Stability

Purdue University, West Lafayette IN

Investigators

Abstract

This project aims to study the way in which infinite-dimensional space can be transformed. The various transformations involved are collectively known as operators. There are some well-behaved subcollections of such operators that first came to prominence as a tool introduced by John von Neumann to further his work in quantum mechanics and that are now known as operator algebras. The principal investigator seeks to collect detailed information about these algebras. Within the class of operator algebras are those that can be obtained by simply moving a geometric shape around, such as rotating a circle. These are known as the algebras associated with dynamical systems. The project will concentrate on this class of algebras. The project aims to study the question of whether the seemingly weak technical condition of strict comparison entails the seemingly stronger condition of Jiang-Su absorption for the operator algebras associated with topological dynamical systems. This would complete a conjecture of Winter and the principal investigator for algebras of this type, a conjecture that has been largely confirmed otherwise. In the absence of Jiang-Su absorption the goal is to show that these algebras do at least possess a less stringent property, namely, that the set of invertible elements of such an algebra constitutes a dense subset of the algebra.