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 Cb(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 Lp(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 Lp-C∗-algebra of G. Using this result, we construct uncountably many new examples of exotic quantum group norms for compact quantum groups.
ASJC Scopus subject areas
- Applied Mathematics