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

Abstract

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