Crossed product splitting of intermediate operator algebras via 2-cocycles

Abstract

We investigate the C*-algebra inclusions $B\subset A{⋊}_{\mathrm{r}}\Gamma$ arising from inclusions $B\subset A$ of $\Gamma$-C*-algebras. The main result shows that, when $B\subset A$ is C*-irreducible in the sense of Rørdam, and is centrally $\Gamma$-free in the sense of the author, then after tensoring with the Cuntz algebra ${\mathcal{O}}_2$, all intermediate C*-algebras $B\subset C\subset A{⋊}_{\mathrm{r}}\Gamma$ enjoy a natural crossed product splitting \[\mathcal{O}_2\otimes C=(\mathcal{O}_2 \otimes D) \rtimes_{{\rm r}, γ, \mathfrak{w}} Λ\] for $D:=C\cap A$, some $\Lambda <\Gamma$, and a subsystem $(\gamma ,\mathfrak{w})$ of a unitary perturbed cocycle action $\Lambda \curvearrowright {\mathcal{O}}_2\otimes A$. As an application, we give a new Galois's type theorem for the Bisch--Haagerup type inclusions \[A^K \subset A\rtimes_{\rm r} Γ\] for actions of compact-by-discrete groups $K⋊\Gamma$ on simple C*-algebras. Due to a K-theoretical obstruction, the operation ${\mathcal{O}}_2\otimes -$ is necessary to obtain the clean splitting. Also, in general 2-cocycles $\mathfrak{w}$ appearing in the splitting cannot be removed even further tensoring with any unital (cocycle) action. We show them by examples, which further show that ${\mathcal{O}}_2$ is a minimal possible choice. We also establish a von Neumann algebra analogue, where ${\mathcal{O}}_2$ is replaced by the type I factor $\mathbb{B}({\ell }^2(\mathbb{N}))$.

Explanations for Remark 5.5 (1) added, other minor revisions, 34 pages, To appear in Mathematische Annalen

References in corpus (3)

• On characterizations of amenable C*-dynamical systems and new examples
• Charmenability of higher rank arithmetic groups
• Noncommutative ergodic theory of higher rank lattices