Operator approximation property for noncommutative $L_p$ spaces of $\mathcal{L}\mathrm{SL}_{2d-1}(\mathbb{Z})$

Determine whether the noncommutative $L_p$ space of $\mathcal{L}\mathrm{SL}_{2d-1}(\mathbb{Z})$ has the operator approximation property for some $p\neq2$ in the range $2d/(d+1)\leq p\leq2d/(d-1)$, thereby resolving the open problem proposed in connection with the Bochner–Riesz and Kakeya conjectures.

Background

The paper cites a problem proposed by M. de la Salle concerning the completely bounded approximation property of noncommutative LpL_p spaces associated with groups related to special linear groups. The relevant exponent interval coincides with the range appearing in the Bochner–Riesz conjecture.

The question is explicitly presented as an open problem and is included as motivation for the broader connections between noncommutative harmonic analysis, approximation properties, and classical restriction-type conjectures.

References

In particular, he proposed the following open problem: does the noncommutative $L_p$ space of $\mathcal{L}\mathrm{SL}_{2d-1}(\mathbb{Z})$ have the OAP for some $p\neq 2$ and $\frac{2d}{d+1}\leq p\leq\frac{2d}{d-1}$.

— Weak type $(1,1)$ boundedness of Bochner--Riesz means at the critical index on quantum tori  (2610.03107 - Lai, 2 Oct 2026) in Section 1, Introduction and main result