Quantum Symmetries and Strong Haagerup Inequalities

In this paper, we consider families of operators ({{x_r}_{r in Lambda}}) in a tracial C*-probability space ({({mathcal{A}}, varphi)}) , whose joint *-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 *-free R-diagonal families obtained by Kemp–Speicher (J Funct Anal 251:141–173, 2007). 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*-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 (Vergnioux in J Oper Theory 57:303–324, 2007).