Strong boundary points in every fiber for Au(Bℓp) beyond p=2

Ascertain whether, for ℓp with 1 ≤ p < ∞ and p ≠ 2, every fiber of the spectrum M(Au(Bℓp)) contains strong boundary points.

Background

The authors recall that strong boundary points and peak points yield singleton Gleason parts and are central in understanding the geometry of spectra. Using an automorphism argument in ℓ2, they show that all fibers in M(Au(Bℓ2)) contain strong boundary points.

They then note the absence of analogous results for other p and explicitly state that it is unknown whether this property extends beyond p=2. Resolving this would clarify the prevalence of strong boundary points across fibers in M(Au(Bℓp)).

References

Therefore, the previous example along with what we have commented in Remark 3.6 allows us to conclude that all the fibers of M(A (Bu)) 2ave strong boundary points. We do not know if the same holds for other values of p.

— Fibers and Gleason parts for the maximal ideal space of $\mathcal A_u(B_{\ell_p})$ (2409.13889 - Dimant et al., 20 Sep 2024) in Section 5 (Final comments and open questions)