C*-irreducible regular inclusions, Galois correspondence and aperiodicity

Abstract

We characterise C*-irreducible regular C*-inclusions using a number of different conditions considered by different authors. In particular, we show that all C*-irreducible regular inclusions $A\subseteq B$ are modelled by outer Fell bundles $(B_{g})_{g\in G}$ over discrete groups with a simple unit fibre $A=B_1$. In this case, we prove a bijection between intermediate C*-algebras $A\subseteq C \subseteq B$ and subgroups $H$ of $G$. This extends the Galois correspondence for reduced crossed products by discrete group actions established by Cameron-Smith. We relate it to the Galois correspondences of Izumi and Mukohara for fixed-point algebras of actions of compact abelian groups, and the mixed inclusion of a fixed-point algebra in a reduced crossed product considered by Echterhoff-Rørdam. In addition, using a recent result of Geffen-Ursu, we show that the inclusion of a fixed-point subalgebra $A\subseteq B$ of an action of $\mathbb{T}$ or $\mathbb{Z}/p$ for a square-free number $p>0$ is aperiodic if and only if $A$ detects ideals in $B$. We apply this to give examples of C*-irreducible inclusions coming from Cuntz-Pimsner algebras, including crossed products by endomorphisms or transfer operators. In particular, we characterise when a core subalgebra of a graph C*-algebra is C*-irreducible. Lastly, we show that a general regular topologically graded C*-inclusion $A\subseteq B$ is aperiodic and has a unique pseudo-expectation provided $A$ detects ideals in all intermediate C*-algebras of $B$. This partially answers a question by Pitts-Zarikian.