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

Let [Figure not available: see fulltext.] be a C*-discrete quantum group and let [Figure not available: see fulltext.] be the discrete quantum group associated with [Figure not available: see fulltext.]. Suppose that there exists a continuous action of [Figure not available: see fulltext.] on a unital C*-algebra [Figure not available: see fulltext.] so that [Figure not available: see fulltext.] becomes a [Figure not available: see fulltext.]-algebra. If there is a faithful irreducible vacuum representation π of [Figure not available: see fulltext.] on a Hilbert space H = [Figure not available: see fulltext.] with a vacuum vector Ω, which gives rise to a [Figure not available: see fulltext.]-invariant state, then there is a unique C*-representation (θ, H) of [Figure not available: see fulltext.] supplemented by the action. The fixed point subspace of [Figure not available: see fulltext.] under the action of [Figure not available: see fulltext.] is exactly the commutant of θ([Figure not available: see fulltext.]).