Non-commutative factors for an irrational rotation of the circle

In a joint work with Tattwamasi Amrutam and Eli Glasner, we study intermediate $C^*$-algebras of the form $C^*_r(\Gamma) < \mathcal{A} < C(X) \rtimes \Gamma$, where $\Gamma \curvearrowright X$ is a given minimal action of a countable discrete group $\Gamma$ on a compact space $X$. Every $\Gamma$-factor of the given topological dynamical system $X \rightarrow Y$ gives rise to an intermediate algebra of the form $\mathcal{A} = C(Y) \rtimes \Gamma$, and by analogy we may think of more general factors as representing ‘‘non-abelian’’ factors. Let us call the dynamical system ``reflecting’’ if the only intermediate algebras come from dynamical factors.

We show that another source of intermediate algebras comes from ideals in $C^*_r(\Gamma)$. In particular, we show that if $\Gamma$ is not $C^*$-simple, $X$ admits a $\Gamma$-invariant probability measure, and the cardinality of $X$ is at least $3$, then the system is not reflecting.

In the talk, I will focus on the example highlighted in the title. In this case, we obtain a complete description of all intermediate algebras in terms of some combinatorial data described in terms of ideals in $C^*_r(\mathbb{Z})$. In particular there are uncountably many intermediate algebras, as compared to only countably many dynamical factors. I will show how our description can often be used in order to obtain structural information about the algebras, such as simplicity, the existence of a center, and a closed formula for the algebra generated by two given ones.