Structure and classification of flows and quantum symmetries on C*-algebras

Objective

Following the enormous progress registered in the Elliott classification programme for simple nuclear C*-algebras, it is an appropriate time to extend our understanding of the symmetries of such C*-algebras. Arguably the biggest achievement obtained so far in this direction is Gabe and Szabó's classification of certain group actions on Kirchberg algebras. However, in the case of the group being the real numbers, their results do not improve Szabó's previous classification of Rokhlin flows on Kirchberg algebras. While classifying actions of discrete groups proved to be a fundamental tool for understanding the structure and symmetries of operator algebras, certain applications to geometry or physics are usually related to time evolutions, or continuous actions of the real numbers. One fundamental feature which makes flows particularly difficult to manoeuvre is the potential presence of KMS states. One goal of this project is to extend and deepen our understanding of flows on classifiable C*-algebras via the study of their associated KMS states. In particular, an important emphasis will be put on taking inspiration from techniques developed in the classification theory of C*-algebras to gain insight of the structure and classification of C*-dynamical systems. In fact, the focus shall be split between periodic flows, which will be viewed as circle actions, and non-periodic ones. The second goal of this proposal is to break new ground into the study of actions of unitary tensor categories (UTC's) on C*-algebras. Given the limited success in classifying actions of general UTC's on C*-algebras, this project proposes a full paradigm-shift. Precisely, the goal is to obtain classification of actions via classifying equivariant morphisms. Although this strategy has been successfully utilised to classify certain group actions on Kirchberg algebras, a UTC version of such a result requires the development of a series of new methods.