New insights into Gleason parts for an algebra of holomorphic functions

We study the structure of the spectrum of the algebra of uniformly continuous holomorphic functions on the unit ball of $\ell_p$. Our main focus is the relationship between \emph{Gleason parts} and \emph{fibers}. For every $z \in B_{\ell_p}$ with $1 < p < \infty$, we prove that the fiber over $z$ contains $2^{\mathfrak{c}}$ distinct Gleason parts. We also investigate some of the properties of these Gleason parts and show the existence of many strong boundary points in certain fibers. We then examine the case $p = 1$, where similar results on the abundance of Gleason parts within the fibers hold, although the arguments required are more involved. Our results extend and complete earlier work on the subject, providing answers to previously posed questions.

💡 Research Summary

This paper presents a detailed study of the structure of the spectrum (maximal ideal space) M(A_u(B_ℓ_p)) of the Banach algebra A_u(B_ℓ_p) of uniformly continuous holomorphic functions on the unit ball of the sequence space ℓ_p, for 1 ≤ p < ∞. The central theme is the intricate relationship between two fundamental ways of partitioning the spectrum: fibers and Gleason parts.

Fibers are defined via the restriction map π(φ) = φ|_ℓ_p’, which projects each multiplicative linear functional φ onto the closed unit ball of the bidual B_ℓ_p’’. For a point z in B_ℓ_p’’, the fiber over z, denoted M_z, is the set π^{-1}(z). Gleason parts, on the other hand, are equivalence classes under the Gleason distance ∥φ - ψ∥ = sup{|φ(f) - ψ(f)| : f ∈ A_u(B_ℓ_p), ∥f∥ ≤ 1}; two homomorphisms belong to the same part if this distance is less than 2, indicating analytic similarity.

For the case 1 < p < ∞, the space ℓ_p is uniformly convex. It is known that for points z on the unit sphere S_ℓ_p, both the fiber M_z and the Gleason part G_P(δ_z) are singletons. Therefore, the interesting structure lies inside the open unit ball B_ℓ_p.

The main result of Section 3 (Theorem 3.2) establishes a striking property for interior points. For any z ∈ B_ℓ_p, let s = (1 - ∥z∥^p)^{1/p}. The authors consider the sequence of evaluation homomorphisms at the points π_n(z) + s e_{n+1}, where π_n(z) is the truncation of z to its first n coordinates and e_{n+1} is the (n+1)-th standard basis vector. They prove that the set of all w*-accumulation points of this sequence has cardinality 2^𝔠 (the maximum possible). Crucially, each such accumulation point φ belongs to a distinct Gleason part. Moreover, they show that the entire Gleason part G_P(φ) is contained within the same fiber M_z. This means the fiber over an interior point not only intersects but fully contains a vast family of 2^𝔠 different Gleason parts, revealing an extremely rich internal structure.

The proof involves several sophisticated steps. It begins with a special case of finitely supported z with rational norm (Proposition 3.3). Then, it is shown that for any z, there exists an element in M_z not in the Gleason part of δ_z (Proposition 3.4). Through a technical construction leveraging properties of homogeneous polynomials and a product norm formula (Lemma 2.4), the authors demonstrate that different accumulation points of the constructed sequence must lie in different Gleason parts, proving the existence of 2^𝔠 parts (Proposition 3.6). Finally, they confirm these parts reside entirely within M_z (Proposition 3.9). Additionally, Proposition 3.12 provides a description of those homomorphisms within these Gleason parts that are limits of evaluations (i.e., not in the corona).

Section 4 addresses the case p = 1. The space ℓ_1 is not uniformly convex, and fibers over points on the unit sphere can be non-singleton, making the analysis more involved. Nevertheless, the authors prove an analogous result (Theorem 4.1): for any z in the closed unit ball of the bidual B_ℓ_1’’, the fiber M_z intersects 2^𝔠 distinct Gleason parts. The arguments required here are different and more complex than for p > 1. The section concludes with Proposition 4.6 and Example 4.7, which identify a subset of the unit sphere of ℓ_1’’ for which the corresponding fibers intersect different Gleason parts.

Overall, this work significantly advances the understanding of the spectrum of algebras of holomorphic functions on infinite-dimensional Banach spaces. It resolves questions posed in earlier literature, particularly regarding the abundance of Gleason parts within fibers for non-integer p, and provides a detailed picture of the interaction between the algebraic (fiber) and analytic (Gleason part) structures in the concrete setting of ℓ_p spaces.