Equivariance and Imprimitivity for Discrete Hopf C*-Coactions

Abstract

Let U, V, and W be multiplicative unitaries coming from discrete Kac systems such that W is an amenable normal submultiplicative unitary of V with quotient U. We define notions for right-Hilbert bimodules of coactions of S_V and (S_V)^, their restrictions to S_W and (S_U)^, their dual coactions, and their full and reduced crossed products. If N(A) denotes the imprimitivity bimodule associated to any coaction of S_V on a C*-algebra A by Ng's imprimitivity theorem, then for any suitably nondegenerate injective coaction of S_V on a right-Hilbert A - B bimodule X we establish an isomorphism between two tensor product bimodules involving N(A), N(B), and certain crossed products of X. This can be interpreted as a natural transformation between two crossed-product functors.