A tracial characterization of Furstenberg's ×p,×q conjecture

Bruce, C. and Scarparo, E. (2024) A tracial characterization of Furstenberg's ×p,×q conjecture. Canadian Mathematical Bulletin = Bulletin canadien de mathématiques, 67(1), pp. 244-256. (doi: 10.4153/S0008439523000693)

Abstract

We investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers p and q, we show that Furstenberg’s xp, xq conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the C∗-algebra of the group Z[1/pq] ⋊ Z2. We also compute the primitive ideal space and K-theory of C∗(Z[1/pq ] ⋊ Z2).

• Item Type: Articles
• Additional Information: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 817597). C. Bruce has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101022531.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bruce, Dr Chris and Scarparo, Dr Eduardo
• Authors: Bruce, C., and Scarparo, E.
• College/School: College of Science and Engineering > School of Mathematics and Statistics
• Journal Name: Canadian Mathematical Bulletin = Bulletin canadien de mathématiques
• Publisher: Cambridge University Press
• ISSN: 0008-4395
• ISSN (Online): 1496-4287
• Published Online: 06 September 2023
• Copyright Holders: Copyright © 2023 The Author(s)
• First Published: First published in Canadian Mathematical Bulletin = Bulletin canadien de mathématiques 67(1): 244-256
• Publisher Policy: Reproduced in accordance with the publisher copyright policy

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
312369		Groups, operator algebras, and dynamical systems	Xin Li	European Commission (EC)	101022531	M&S - Mathematics