Quantum Symmetries and Strong Haagerup Inequalities

Abstract

In this paper, we consider families of operators $\{x_r{\}}_{r\in \Lambda }$ in a tracial C${}^{\ast }$-probability space $(\mathcal{A},\varphi )$, whose joint $\ast$-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups $\{H_n^{+}{\}}_{n\in \mathbb{N}}$. We prove a strong form of Haagerup's inequality for the non-self-adjoint operator algebra $\mathcal{B}$ generated by $\{x_r{\}}_{r\in \Lambda }$, which generalizes the strong Haagerup inequalities for $\ast$-free R-diagonal families obtained by Kemp-Speicher \cite{KeSp}. As an application of our result, we show that $\mathcal{B}$ always has the metric approximation property (MAP). We also apply our techniques to study the reduced C${}^{\ast }$-algebra of the free unitary quantum group $U_n^{+}$. We show that the non-self-adjoint subalgebra ${\mathcal{B}}_n$ generated by the matrix elements of the fundamental corepresentation of $U_n^{+}$ has the MAP. Additionally, we prove a strong Haagerup inequality for ${\mathcal{B}}_n$, which improves on the estimates given by Vergnioux's property RD \cite{Ve}.

References in corpus (3)

• A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation
• De Finetti theorems for easy quantum groups
• Strong Haagerup inequality with operator coefficients

Cited by in corpus (2)

• Lacunary Fourier series for compact quantum groups
• Hardy-Littlewood inequalities on compact quantum groups of Kac type