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A tracial characterization of Furstenberg’s $\times p,\times q$ conjecture

Part of: Selfadjoint operator algebras Ergodic theory

Published online by Cambridge University Press:  06 September 2023

Chris Bruce Open the ORCID record for Chris Bruce [Opens in a new window]  and
Eduardo Scarparo Open the ORCID record for Eduardo Scarparo [Opens in a new window]
Chris Bruce*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom
Eduardo Scarparo
Affiliation:
Center for Engineering, Federal University of Pelotas, Pelotas, Brazil e-mail: eduardo.scarparo@ufpel.edu.br
*
e-mail: Chris.Bruce@glasgow.ac.uk
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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 $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the $C^*$-algebra of the group $\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2$. We also compute the primitive ideal space and K-theory of $C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$.

Keywords

Furstenberg conjecturetracial statesprimitive idealsK-theory

MSC classification

Primary: 37A55: Relations with the theory of $C^*$-algebras 46L05: General theory of $C^*$-algebras

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 1 , March 2024 , pp. 244 - 256
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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