[Paper Review] $L_p$-Representations of Discrete Quantum Groups

Michael Brannan, Zhong‐Jin Ruan|arXiv (Cornell University)|Apr 16, 2014

Advanced Operator Algebra Research31 references6 citations

TL;DR

This paper introduces $L_p$-representations and $D$-C$^*$-algebras for discrete quantum groups, generalizing group C$^*$-algebra completions via subspaces of the multiplier algebra. It establishes that for unimodular discrete quantum groups like the free unitary quantum groups $\mathbb{F}U_N$, the $L_p$-C$^*$-algebras $C^*_p(\mathbb{F}U_N)$ for $p \in (2,\infty)$ are exotic, non-isomorphic to both the universal and reduced C$^*$-algebras, and admit no characters, proving their non-simplicity and structural novelty.

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.

Motivation & Objective

• To extend the theory of $D$-C$^*$-algebras from discrete groups to locally compact quantum groups, particularly focusing on $L_p$-completions of $L_1(\mathbb{G})$.
• To characterize positive definite functions on unimodular orthogonal and unitary free quantum groups that extend to states on $L_p$-C$^*$-algebras.
• To construct uncountably many new exotic quantum group norms for compact quantum groups via $L_p$-completions of discrete quantum groups.
• To establish the non-isomorphism and non-simplicity of $C^*_p(\mathbb{F}U_N)$ for $p>2$, distinguishing them from universal and reduced C$^*$-algebras.

Proposed method

• Define $D$-representations of $L_1(\mathbb{G})$ as bounded representations where coefficient functions lie in a subspace $D \subset C_b(\mathbb{G})$.
• Construct a $C^*$-norm $\|\omega\|_D = \sup_\pi \|\pi(\omega)\|$ over all $D$-representations $\pi$, leading to the $D$-C$^*$-algebra $C^*_D(\mathbb{G})$.
• Use Pontryagin duality to interpret $C^*_D(\mathbb{G})$ as a completion of $\text{Pol}(\hat{\mathbb{G}})$, enabling construction of exotic norms for compact quantum groups.
• Apply the $\ell_q$-property of rapid decay to characterize weak $L_q$-boundedness of positive definite functions, linking it to $L_p$-completions.
• Construct a net of norm-one positive definite functions $\tilde{\psi}_r$ on $\mathbb{F}U_N$ using projections onto quantum subgroups, and analyze their $L_p$-boundedness via associated orthogonal quantum group functions.
• Use divergence of $\sum_g r^{\ell(g)p} \dim(H_g)^2$ for $r \to 1^-$ to prove no characters exist on $C^*_p(\mathbb{F}U_N)$, implying non-simplicity.

Experimental results

Research questions

• RQ1 Which positive definite functions on unimodular free quantum groups extend to states on $L_p$-C$^*$-algebras?
• RQ2 Are the $L_p$-C$^*$-algebras $C^*_p(\mathbb{F}U_N)$ for $p>2$ exotic, i.e., strictly intermediate between the universal and reduced C$^*$-algebras?
• RQ3 Are the $C^*_p(\mathbb{F}U_N)$ algebras non-isomorphic to each other and to $C_u(U_N^+)$ and $C(U_N^+)$ for $p>2$?
• RQ4 Do the $C^*_p(\mathbb{F}U_N)$ algebras admit traces, and is the trace unique?

Key findings

• For each $p \in (2,\infty)$, the $L_p$-C$^*$-algebra $C^*_p(\mathbb{F}U_N)$ is not isomorphic to the reduced C$^*$-algebra $C(U_N^+)$, due to non-simplicity and the absence of characters.
• The canonical quotient map $C^*_{p'}(\mathbb{F}U_N) \to C^*_p(\mathbb{F}U_N)$ is not injective for $2 \leq p < p' < \infty$, proving the algebras are exotic.
• The $C^*_p(\mathbb{F}U_N)$ algebras have no characters, as no weakly $L_p$ positive definite function can satisfy $\sum_g r^{\ell(g)p} \dim(H_g)^2 < \infty$ for $r$ near 1.
• The $L_p$-completions yield uncountably many new exotic quantum group norms for compact quantum groups $U_N^+$ via duality.
• The characterization of $L_p$-bounded positive definite functions on $\mathbb{F}U_N$ relies on the $\ell_q$-property of rapid decay and the behavior of associated Chebyshev polynomials $S_{2n}(rN)$.
• The non-isomorphism of $C^*_p(\mathbb{F}U_N)$ for different $p>2$ follows from the non-extendability of certain positive definite functions to $L_p$-completions.