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arXiv · 2304.04456

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

Chris Bruce·Eduardo Scarparo
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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 $C^*(\mathbb{Z}[\frac{1}{pq}]\rtimes\mathbb{Z}^2)$.

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Chris Bruce, Eduardo Scarparo. 2023-04-10. A tracial characterization of Furstenberg's $\times p,\times q$ conjecture. https://doi.org/10.4153/s0008439523000693

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