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A tracial characterization of Furstenberg’s 
$\times p,\times q$ conjecture
Published online by Cambridge University Press: 06 September 2023
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 
$C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$.
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society
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