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A tracial characterization of Furstenberg’s
$\times p,\times q$ conjecture
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 67 / Issue 1 / March 2024
- Published online by Cambridge University Press:
- 06 September 2023, pp. 244-256
- Print publication:
- March 2024
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- Article
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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 
$C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$.
