$\mathrm C^*$-Homomorphisms and duality of $\mathrm C^*$-discrete quantum groups

Let $\mathscr D$ be a $\mathrm C^*$-discrete quantum group and let $\mathscr D_0$ be the discrete quantum group associated with $\mathscr D$. Suppose that there exists a continuous action of $\mathscr D$ on a unital $\mathrm C^*$-algebra $\mathscr A$ so that $\mathscr A$ becomes a $\mathscr D$-algebra. If there is a faithful irreducible vacuum representation $\pi$ of $\mathscr A$ on a Hilbert space $H=\mathscr A$ with a vacuum vector $\Omega$, which gives rise to a $\mathscr D$-invariant state, then there is a unique $\mathrm C^*$-representation $(\theta,H)$ of $\mathscr D$ supplemented by the action. The fixed point subspace of $\mathscr A$ under the action of $\mathscr D$ is exactly the commutant of $\theta(\mathscr D)$.