Distinct morphisms sharing GP(δ0) in interior fibers for Au(Bℓp)

Show the existence or nonexistence of distinct morphisms ϕ ≠ ψ, lying in fibers over interior points of M(Au(Bℓp)) for 1 ≤ p < ∞, such that GP(ϕ) = GP(ψ) = GP(δ0).

Background

Known constructions (e.g., analytic disks) produce elements of fibers over interior points that lie in GP(δ0), and various results place substantial analytic structure within GP(δ0). However, whether two distinct homomorphisms from the same interior fiber can share exactly the Gleason part GP(δ0) remains open.

Resolving this would clarify the structure and multiplicity of Gleason parts intersecting nontrivial fibers across ℓp spaces.

References

Open problem 3. For M(A (B )), 1 ≤ p < ∞, show the existence (or lack) of morphisms ϕ = ψ, lying in fibers over interior points, such that GP(ϕ) = GP(ψ) = GP(δ ).

— 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)