Amenability, structure and regularity of group actions on C*-algebras

• Applicant: Eusebio Gardella, Ph.D.
• Subject Area: Mathematics
• Term: from 2018 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 418366465

In mathematics, the study of symmetries and group actions is one of the most fundamental and prolific fields of research. In operator algebras, the classification of integer actions was instrumental in Connes' award-winning proof of uniqueness of amenable type III factors. Far reaching generalizations culminated in the classification of amenable group actions on amenable factors. Despite the efforts, the area of C*-dynamics is far less developed than its von Neumann algebraic counterpart. However, recent breakthroughs have given new impetus to the field. This project will make exceptional progress in C*-dynamics, consolidating the area as one of the most active ones in C*-algebras. Along with the main goals of this project, we will also explore a number of problems related to dynamics both on C*-algebras and von Neumann algebras, whose solution will improve our understanding of the field. Some of these problems have tight connections to topological and measurable dynamics, and are thus of an interdisciplinary nature.The project contains three fundamental Research Units, with ambitious and promising goals. In Unit A, we will clarify the relationship between amenability of a group and cocycle-rigidity of its actions. We expect actions of amenable groups to be considerably more tractable, suggesting that their classification ought to be possible. A crucial step, to be carried out in Unit B, is constructing a model action of any amenable group on the Jiang-Su algebra, similarly to what was done for the hyperfinite II-1 factor. Such a model is likely to play a central role in our setting as well. Concretely, absorption of the model action is expected to be equivalent to a number of naturally occurring freeness- and Rokhlin-type properties; this will be explored in Unit C. Establishing these facts will be key in showing that actions absorbing the model action can be classified. A distinguishing feature of our project is the large extent to which ideas and techniques from von Neumann algebras will be combined in novel ways with tools in C*-algebras, to make outstanding advances in the study of group actions. Nowadays, the fields of C*-algebras and von Neumann algebras have become so specialized that only few researchers work in their intersection. However, it has recently become clear that both areas would profit from closer interactions. For instance, many significant advances in the structure theory of nuclear C*-algebras relied on techniques developed specifically in the setting of amenable factors. There are also a number of instances where C*-algebraic methods have proved to be useful in von Neumann algebras. The theories of C*-algebras and von Neumann algebras are intimately intertwined, and this project aims at exploiting their interdependence.

• DFG Programme: Research Grants