C*-Homomorphisms and duality of C*-discrete quantum groups

Abstract

Let D be a C * -discrete quantum group and let D 0 be the discrete quantum group associated with D. Suppose that there exists a continuous action of D on a unital C * -algebra A so that A becomes a D-algebra. If there is a faithful irreducible vacuum representation π of A on a Hilbert space H = (π(A )Ω) with a vacuum vector Ω, which gives rise to a D-invariant state, then there is a unique C * -representation (θ, H) of D supplemented by the action. The fixed point subspace of A under the action of D is exactly the commutant of θ(D).