Representation and duality of unimodular C*-discrete quantum groups

Suppose that D is a C*-discrete quantum group and D₀ a discrete quantum group associated with D. If there exists a continuous action of D on an operator algebra L(H) so that L(H) becomes a D-module algebra, and if the inner product on the Hilbert space H is D-invariant, there is a unique C*-representation θ of D associated with the action. The fixed-point subspace under the action of D is a Von Neumann algebra, and furthermore, it is the commutant of θ(D) in L(H).