Irreducible Inclusions of Factors, Multiplicative Unitaries, and Kac Algebras

From an irreducible depth 2 inclusion of factors, verifying a regularity condition, we construct a multiplicative unitary, and an action, at every level of the canonical tower constructed from the inclusion; when this inclusion admits a faithful semi-finite normal operator-valued weight, stronger conditions are given, and the tower appears then as a crossed-product construction. In particular we rederive Herman and Ocneanuʹs results when the inclusion admits a faithful normal conditional expectation, and the tower is then the crossed-product construction, alternatively by a compact quantum group and by its dual, and, more precisely, according to Yamagamiʹs result, by a compact type Kac algebra and by its dual.