Fibers and Gleason parts for the maximal ideal space of \({\mathcal {A}}_u(B_{\ell _p})\)

In the early nineties, R. M. Aron, B. Cole, T. W. Gamelin and W. B. Johnson initiated the study of the maximal ideal space (spectrum) of Banach algebras of holomorphic functions defined on the open unit ball of an infinite dimensional complex Banach space. Within this framework, we investigate the fibers and Gleason parts of the spectrum of the algebra of holomorphic and uniformly continuous functions on the unit ball of \(\ell _p\) ${\ell }_p$ ( \(1\le p<\infty \) $1\le p<\infty$ ). We show that the inherent geometry of these spaces provides a fundamental ingredient for our results. We prove that whenever \(p\in {\mathbb {N}}\) $p\in \mathbb{N}$ ( \(p\ge 2\) $p\ge 2$ ), the fiber of every \(z\in B_{\ell _p}\) $z\in B_{{\ell }_p}$ contains a set of cardinal \(2^{{\mathfrak {c}}}\) $2^{\mathfrak{c}}$ such that any two elements of this set belong to different Gleason parts. For the case \(p=1\) $p=1$ , we complete the known description of the fibers, showing that, for each \(z\in \overline{B}_{\ell _1''}{\setminus } S_{\ell _1}\) $z\in {\overline{B}}_{{\ell }_1^{\prime \prime }}\backslash S_{{\ell }_1}$ , the fiber over z is not a singleton. Also, we establish that different fibers over elements in \(S_{\ell _1''}\) $S_{{\ell }_1^{\prime \prime }}$ cannot share Gleason parts.