Gleason part GP(δz) in non-extreme fibers for Au(Bℓ1)

Determine whether, for the spectrum M(Au(Bℓ1)), given z ∈ Bℓ1′ with z ∈ Sℓ1 and z not a real extreme point of the unit ball of c0, there exists a homomorphism ϕ in the fiber Mz such that GP(ϕ) = GP(δz).

Background

In the ℓ1 case, the authors complete aspects of the description of fibers over points in the bidual and analyze interactions with Gleason parts. They show non-singleton fibers over many points and establish certain inclusions of disks in Gleason parts. Nonetheless, whether the Gleason part of δz is met by another homomorphism from the same fiber remains unresolved when z lies in the canonical copy Sℓ1 but is not a real extreme point of Bc0.

This question targets the fine structure of Gleason parts intersecting fibers over interior points in the non-reflexive ℓ1 setting.

References

Open problem 2. For M(A (B u ℓ1), given z ∈ B ℓ1′such that z ∈ S ℓ1 and z is not a real extreme point of B c0, decide whether there exists ϕ ∈ M witz GP(ϕ) = GP(δ ). z

— 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)