L_p-representations of discrete quantum groups

Given a locally compact quantum group G, we define and study representations and C∗-completions of the convolution algebra L₁(G) associated with various linear subspaces of the multiplier algebra C_b(G). For discrete quantum groups G, we investigate the left regular representation, amenability and the Haagerup property in this framework. When G is unimodular and discrete, we study in detail the C∗-completions of L₁(G) associated with the non-commutative Lp-spaces L_p(G). As an application of this theory, we characterize (for each p ∈ [1,∞]) the positive definite functions on unimodular orthogonal and unitary free quantum groups G that extend to states on the L_p-C∗-algebra of G. Using this result, we construct uncountably many new examples of exotic quantum group norms for compact quantum groups.