Abstract
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).
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Communicated by Y. Kawahigashi
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Brannan, M. Quantum Symmetries and Strong Haagerup Inequalities. Commun. Math. Phys. 311, 21–53 (2012). https://doi.org/10.1007/s00220-012-1447-6
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DOI: https://doi.org/10.1007/s00220-012-1447-6