Groups, operator algebras, and dynamics

Project Information

GOADS

Grant agreement ID: 101022531

DOI

10.3030/101022531

Project closed

EC signature date 24 March 2021

Start date 1 May 2022

End date 30 April 2024

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 212 933,76

EU contribution

€ 212 933,76

Coordinated by

UNIVERSITY OF GLASGOW
United Kingdom

Periodic Reporting for period 1 - GOADS (Groups, operator algebras, and dynamics)

Reporting period: 2022-05-01 to 2024-04-30

Summary of the context and overall objectives of the project

The project lies at the intersection of operator algebras, (semi)group theory, and topological dynamics. We initiated a systematic study of C*-algebras and groupoids attached to algebraic actions of semigroups, with applications to group theory. Such actions are the “irreversible’’ analogues of the well-studied algebraic actions of groups. The project aim comprised two objectives: Objective A was concerned with the structure of the canonical concrete C*-algebras associated with algebraic actions; Objective B was concerned with groupoids associated with algebraic actions and their topological full groups.

This MSCA-IF facilitated many collaborations and interactions of the ER with researcher from around the globe, and it helped the ER obtain a permanent academic position.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

This project resulted in eight deliverables, each in the form of a preprint. Four of these have already been published. Moreover, there is a ninth preprint near completion. This is twice the number of papers predicted in the proposal.

The results have been widely disseminated: The ER has presented at the following 7 conferences/workshops during the fellowship:

7) Workshop "Ideal Structure of C*-algebras from Dynamics and Groups" Erlangen, Germany.
6) Workshop “Groups and Group Dynamics” at the 2023 Thematic Program on Operator Algebras and Applications at the Fields Institute in Toronto, Canada.
5) Workshop titled “Algebra, Geometry and C*-algebras”, International Centre for Mathematical Sciences (ICMS), Edinburgh, UK.
4) Conference “Dilation and classification in operator algebra theory” at the University of Copenhagen, Copenhagen, Denmark.
3) Noncommutative Analysis at the Technion at Technion – Israel Institute of Technology, Haifa, Israel.
2) Canadian Operator Theory Symposium (COSy) in Ottawa, Canada.
1) Workshop C*-algebras and geometry of groups and semigroups, University of Oslo, Oslo, Norway.

Additionally, the ER has given seminar talks in Odense, Cardiff, Warwick, Newcastle, Glasgow, Münster, RIMS Kyoto (x2), and at the Fields Institute.

During this project, the ER co-organised the Glasgow Analysis Seminar for the first 16 months of the project, and co-supervised an MSci student at the University of Glasgow, along with two EPSRC-funded undergraduate students. He also co-organised the “Glasgow Late August Symbolic Dynamics, Groups, and Operators Workshop” at the University of Glasgow.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

That the project advanced the state-of-the-art is witnessed by, for instance, the following deliverables of the project:

(1) Algebraic actions I. C*-algebras and groupoids. With X. Li. Published in the Journal of Functional Analysis.

“We associated an ample groupoid to each algebraic action of a semigroup, and characterise properties of our groupoid in terms of the initial algebraic action. Then, we use these groupoids to obtain new structural results for the concrete C*-algebras associated with algebraic actions, which were out of reach using existing techniques.”

(2) Algebraic actions II. Groupoid rigidity. With X. Li. Submitted preprint.

“We studied groupoid rigidity for the groupoids we defined previously. Surprising, for examples from algebraic number theory, our groupoids demonstrate complete rigidity in that two such groupoids are isomorphic if and only if the underlying number-theoretic data is isomorphic.”

(3) Groupoid homology and K-theory for algebraic actions. With Y. Kubota and T. Takeishi. In preparation.

“We analyse groupoid homology for the ample groupoids that X. Li and the ER associated with algebraic actions. For examples coming from rings of algebraic integers or from integral dynamics (in the sense of Barlak—Omland—Stammeier), we are able to completely compute all homology groups. This leads to, for instance, structural results for topological full groups of number-theoretic origin.”

(4) Normal coactions extend to the C*-envelope. With K.A. Brix and A. Dor-On. Submitted preprint.

``We resolved an open problem considered by Kakariadis, Katsoulis, Laca, and X. Li, and provides an elementary proof of a prominent result of Sehnem. As an application, we resolve a question of Li by identifying the C*-envelope of the operator algebra arising from a groupoid-embeddable category and of cancellative right LCM monoids. This latter class includes many examples of monoids that are not group-embeddable. Our methods rely on groupoid-techniques, in which the ER had become an expert.”