Crossed product splitting of intermediate operator algebras via 2-cocycles

We investigate the C*-algebra inclusions $B\subset A{⋊}_{\mathrm{r}}\mathrm{\Gamma }$ arising from inclusions $B\subset A$ of $\mathrm{\Gamma }$-C*-algebras. The main result shows that, when $B\subset A$ is C*-irreducible in the sense of Rørdam, and is centrally $\mathrm{\Gamma }$-free in the sense of the author, then after tensoring with the Cuntz algebra ${\mathcal{O}}_2$, all intermediate C*-algebras $B\subset C\subset A{⋊}_{\mathrm{r}}\mathrm{\Gamma }$ enjoy a natural crossed product splitting $${\mathcal{O}}_2\otimes C=({\mathcal{O}}_2\otimes D){⋊}_{\mathrm{r},\gamma ,\mathfrak{w}}\mathrm{\Lambda }$$ for $D:=C\cap A$, some $\mathrm{\Lambda }<\mathrm{\Gamma }$, and a subsystem $(\gamma ,\mathfrak{w})$ of a unitary perturbed cocycle action $\mathrm{\Lambda }\curvearrowright {\mathcal{O}}_2\otimes A$. As an application, we give a new Galois’s type theorem for the Bisch–Haagerup type inclusions $$A^K\subset A{⋊}_{\mathrm{r}}\mathrm{\Gamma }$$ for actions of compact-by-discrete groups $K⋊\mathrm{\Gamma }$ 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 $\mathbb{B}({\mathrm{\ell }}^2(\mathbb{N}))$.