Crossed product splitting of intermediate operator algebras via 2-cocycles

We investigate the C*-algebra inclusions B ⊂ A ᵣ Γ arising from inclusions B ⊂ A of Γ-C*-algebras. The main result shows that, when B ⊂ A is C*-irreducible in the sense of R{}rdam, and is centrally Γ-free in the sense of the author, then after tensoring with the Cuntz algebra O₂, all intermediate C*-algebras B ⊂ C⊂ A ᵣ Γ enjoy a natural crossed product splitting \[O_2⊗ C=(O_2 ⊗ D) _{{ r}, γ, w} Λ\] for D:= C ∩ A, some Λ<Γ, and a subsystem (γ, w) of a unitary perturbed cocycle action Λ O₂⊗ A. As an application, we give a new Galois's type theorem for the Bisch--Haagerup type inclusions \[A^K ⊂ Aᵣ Γ\] for actions of compact-by-discrete groups K Γ on simple C*-algebras. Due to a K-theoretical obstruction, the operation O₂⊗ - is necessary to obtain the clean splitting. Also, in general 2-cocycles 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 O₂ is a minimal possible choice. We also establish a von Neumann algebra analogue, where O₂ is replaced by the type I factor B(²(N)).