A tracial characterization of Furstenberg’s conjecture

Abstract

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 $C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$ .