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Structure and classification of C*-dynamics

Periodic Reporting for period 1 - SCCD (Structure and classification of C*-dynamics)

Reporting period: 2018-01-01 to 2019-12-31

The research outcome concerns the structure theory of C*-algebras, which is a branch of functional analysis. The focus lies on noncommutative dynamical systems, or also called C*-dynamics, which can for example be a mathematical description of a time evolution in a physical system. Of particular interest are dynamical structures on simple nuclear C*-algebras, which can be thought of as noncommutative topological spaces that are indecomposable into smaller pieces. The classification theory for the underlying objects has made big steps in recent years, and the research carried out over this project capitalizes on the new techniques to study dynamics on them, which can be thought of as generalized symmetries. The main objectives can be summarized as follows:
1) To test a potential classification theory for actions of amenable groups on the class of strongly self-absorbing C*-algebras, which are particularly rigid in nature.
2) Investigate the fine structure group actions on purely infinite C*-algebras, which can be thought of as the well-understood low-dimensional noncommutative spaces.
3) Classify time evolutions, or flows, on classifiable classes of C*-algebras. Of particular interest are flows with the so-called Rokhlin property a la Kishimoto.
In the context of 1), it is shown that every amenable group inside a large natural class admits a unique action on a strongly self-absorbing C*-algebra as long as it has an external origin, or also called strongly outer. The latter criterion is a well-known necessary criterion for such a rigidity phenomon. Furthermore, strongly outer actions of such groups on classifiable C*-algebras automatically absorbs certain model action tensorially as long as a minimal requirement about the underlying C*-algebra is satisfied.

The results obtained for 2) are particularly strong, giving convincing evidence for the possibility of a general classification theory for actions of all amenable groups.

The central outcome of 3) is a satisfactory classification theory for Rokhlin flows, which includes a positive solution to a long-standing conjecture in the field due to Akitaka Kishimoto.
The research methology developed over the project go beyond the state-of-the-art and employ novel ideas in the area of C*-algebras. The results have impact in the research area of operator algebras, which is an important field of mathematics having a multitude of connections to other areas of mathematics and physics. One of the main contributions is the first satisfactory and abstract classification theory for time evolutions on C*-algebras subject to some natural conditions, which opens the door for a subarea of research in C*-algebra theory that had been deemed inaccessible in the past. Adding to this the structural results obtained for actions of discrete amenable groups from the point of view of tensorial absorption, the outcomes of the fellowship have a foreseeable high impact regarding future research on C*-dynamical systems as a whole.