L p {L_{p}} -representations of discrete quantum groups

Given a locally compact quantum group {\mathbb{G}} , we define and study representations and {\mathrm{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 {\mathrm{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}} - {\mathrm{C}^{\ast}} -algebra of {\mathbb{G}} . Using this result, we construct uncountably many new examples of exotic quantum group norms for compact quantum groups.