Proposition 3.5 beyond integer p

Determine whether the conclusion of Proposition 3.5 holds for M(Au(Bℓp)) when p is not an integer; specifically, ascertain if every fiber over a point z ∈ Bℓp contains a set of cardinality 2^ℵ0 with any two elements lying in different Gleason parts, without assuming p ∈ ℕ.

Background

Proposition 3.5 establishes, for integer p ≥ 2, that each fiber over points in Bℓp contains a set of cardinality continuum whose elements lie in pairwise different Gleason parts. The proof relies on the availability of specific polynomials associated with integer degrees.

The authors note this integer requirement is technical and ask whether the structure theorem extends to non-integer p, which would significantly broaden the understanding of Gleason-part partitions within fibers for general ℓp spaces.