Crossed product splitting of intermediate operator algebras via 2-cocycles

We investigate the C*-algebra inclusions B A ₑ arising from inclusions B A of -C*-algebras. The main result shows that, when B A is C*-irreducible in the sense of Rrdam, and is centrally -free in the sense of the author, then after tensoring with the Cuntz algebra O₂, all intermediate C*-algebras B C A ₑ enjoy a natural crossed product splitting ₂ C= (O₂ D) ₑ, , w \ for D: = C A, some <, and a subsystem (, w) of a unitary perturbed cocycle action O₂ A. As an application, we give a new Galois's type theorem for the Bisch--Haagerup type inclusions K A ₑ \ for actions of compact-by-discrete groups K 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) ).