Quantum Symmetries and Strong Haagerup Inequalities

TL;DR

This paper establishes strong Haagerup inequalities for operator algebras invariant under quantum symmetries, demonstrating the metric approximation property and improving bounds for free unitary quantum groups.

Contribution

It generalizes Haagerup inequalities to new classes of operator algebras with quantum symmetry invariance, and applies these results to free unitary quantum groups.

Findings

Proves strong Haagerup inequalities for quantum-invariant operator families.

Shows the generated algebras have the metric approximation property.

Improves bounds for the reduced C*-algebra of free unitary quantum groups.

Abstract

In this paper, we consider families of operators $\{x_r\}_{r \in \Lambda}$ in a tracial C$^\ast$-probability space $(\mathcal A, \phi)$, whose joint $\ast$-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups $\{H_n^+\}_{n \in \N}$. 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}.