Amenable C*-dynamics and their classification

Introduction

Our understanding of amenable C*-algebras has drastically improved over the past decade due to substantial advances in the Elliott classification program. The proposed research is intended to spearhead the classification and structure theory for C*-dynamics, which encompasses groups acting continuously on C*-algebras and the question to which extent their detailed structure is encoded at the level of computable invariants.

Objectives

The objectives involve the following lines of investigation:

1. Discrete Groups Acting on Simple Amenable C-Algebras*
   The case of discrete groups acting on simple amenable C*-algebras, the latter of which have become accessible due to recent methodological breakthroughs within Elliott's program. The task is to develop a dynamical version of Elliott's program, i.e., to consider natural categories of such group actions, identify functorial invariants such as K-theory and traces, and show that these invariants completely classify a naturally occurring category up to a suitable notion of isomorphism.

2. Theory of Invariants for Noncommutative C-Dynamics*
   This field is faced with numerous open problems requiring solutions before a fully general classification theory can be developed. This part of the project addresses the need for effective computational methods for equivariant Kasparov classes, or systematic techniques to exploit equivariant Ext-theory for the purpose of classification.

3. Noncommutative Time Evolutions
   The case of noncommutative time evolutions, which includes new rigidity phenomena for the passage of time in simple C*-algebras, both with and without the presence of equilibrium states.

Thematic Focus

A deeply rooted theme within all these objectives is the presence of amenability for the actions and the C*-algebras, which gives rise to radically new phenomena in the context of non-amenable acting groups. Drawing ideas from diverse areas such as the Elliott program, equivariant Kasparov theory, classical and noncommutative ergodic theory, novel methods are proposed to achieve groundbreaking results in the area of C*-dynamics.