$L_p$-Representations of Discrete Quantum Groups

Given a locally compact quantum group $\mathbb G$, we define and study representations and C$^\ast$-completions of the convolution algebra $L_1(\mathbb G)$ associated with various linear subspaces of the multiplier algebra $C_b(\mathbb G)$. For discrete quantum groups $\mathbb G$, we investigate the left regular representation, amenability and the Haagerup property in this framework. When $\mathbb G$ is unimodular and discrete, we study in detail the C$^\ast$-completions of $L_1(\mathbb G)$ associated with the non-commutative $L_p$-spaces $L_p(\mathbb G)$. As an application of this theory, we characterize (for each $p \in [1,\infty)$) the positive definite functions on unimodular orthogonal and unitary free quantum groups $\mathbb G$ that extend to states on the $L_p$-C$^\ast$-algebra of $\mathbb G$. Using this result, we construct uncountably many new examples of exotic quantum group norms for compact quantum groups.