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

Abstract

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