$L_p$-Representations of Discrete Quantum Groups

Abstract

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.

Major improvements: More results in the locally compact case, and new results obtained for the $L_p$-C${}^{\ast }$-algebras of unitary free quantum groups

References in corpus (3)

• The Property of Rapid Decay for Discrete Quantum Groups
• Orientation of quantum Cayley trees and applications
• K-amenability for amalgamated free products of amenable discrete quantum groups