Ergodic actions of SμU(2)SμU(2) on C∗C∗-algebras from II1II1 subfactors

To a proper inclusion N⊂MN⊂M of II1 factors of finite Jones index [M:N][M:N], we associate an ergodic C∗C∗-action of the quantum group SμU(2)SμU(2) (or more generally of certain groups Ao(F)Ao(F)). The higher relative commutant N′∩Mr−1N′∩Mr−1 can be identified with the spectral space of the rrth tensor power u⊗ru⊗r of the defining representation of the quantum group. The index and the deformation parameter are related by −1≤μ<0−1≤μ<0 and [M:N]=|μ+μ−1|[M:N]=|μ+μ−1|.This ergodic action may be thought of as a virtual subgroup of SμU(2)SμU(2) in the sense of Mackey arising from the tensor category generated by the NN-bimodule NMNNMN. μμ is negative as NMNNMN is a real bimodule.