Corona theorem for Au(BX) in infinite-dimensional Banach spaces (including ℓp)

Determine whether the Corona theorem holds for the algebra Au(BX) on general infinite-dimensional Banach spaces X, and in particular for X = ℓp with 1 ≤ p < ∞; equivalently, decide whether the set of evaluation homomorphisms {δz : z ∈ BX′} is dense in the maximal ideal space M(Au(BX)).

Background

The Corona theorem asserts, loosely speaking, that evaluation homomorphisms are dense in the spectrum of a function algebra. While the classical result of Carleson establishes the theorem for H(D) and it trivially holds for Au(BX) in finite dimensions, the situation in infinite-dimensional settings is significantly more subtle.

In the present work, the authors highlight that determining the validity of the Corona theorem for Au(BX) remains unresolved in general, and specifically for the classical sequence spaces ℓp (1 ≤ p < ∞). This question connects to broader structural properties of spectra, cluster sets, and the richness of fibers in infinite-dimensional holomorphic function algebras.

References

For finite-dimensional Banach spaces, the Corona theorem trivially holds for A uB )X but it is an open question for general infinite dimensional ones, even for the classical p -spaces, 1 ≤ 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 1 (Introduction)