Journal of the
Korean Mathematical Society JKMS

Suppose that $\mathcal D$ is a $C^*$-discrete quantum group and ${\mathcal D}_0$ a discrete quantum group associated with $\mathcal D$. If there exists a continuous action of $\mathcal D$ on an operator algebra $L(H)$ so that $L(H)$ becomes a $\mathcal D$-module algebra, and if the inner product on the Hilbert space $H$ is $\mathcal D$-invariant, there is a unique $C^*$-representation $\theta$ of $\mathcal D$ associated with the action. The fixed-point subspace under the action of $\mathcal D$ is a Von Neumann algebra, and furthermore, it is the commutant of $\theta ({\mathcal D})$ in $L(H)$.

Keywords: discrete quantum group, $C^*$-algebra, representation, duality

MSC numbers: 46K10, 16W30