Strong Haagerup inequalities on non-Kac free orthogonal quantum groups

TL;DR

This paper establishes strong Haagerup inequalities for non-Kac free orthogonal quantum groups, demonstrating their optimality and applying these results to analyze heat semigroup properties and interpolation spaces.

Contribution

It introduces and proves optimal strong Haagerup inequalities for non-Kac free orthogonal quantum groups, addressing challenges due to their non-tracial structure.

Findings

Proved optimal strong Haagerup inequalities for $O_F^+$

Computed the optimal time for ultracontractivity of the heat semigroup

Distinguished between complex and real interpolation spaces on $O_F^+$

Abstract

We present natural analogues of strong Haagerup inequalities on non-Kac free orthogonal quantum groups $O_F^+$ in which $L^p$-analytic problems are harder due to their non-tracial nature. Furthermore, we prove optimality of the inequalities, and apply the obtained results to compute the optimal time for ultracontractivity of the heat semigroup and to distinguish the complex interpolation space $L^p(O_F^+)$ and the real interpolation space $L^{p,p}(O_F^+)$.