Analytic Structure in Fibers

Abstract

Let $B_X$ be the open unit ball of a complex Banach space $X$, and let $\mathcal{H}^\infty(B_X)$ and $\mathcal{A}_u(B_X)$ be, respectively, the algebra of bounded holomorphic functions on $B_X$ and the subalgebra of uniformly continuous holomorphic functions on $B_X$. In this paper we study the analytic structure of fibers in the spectrum of these two algebras. For the case of $\mathcal{H}^\infty(B_X)$, we prove that the fiber in $\mathcal{M}(\mathcal{H}^\infty (B_{c_0}))$ over any point of the distinguished boundary of the closed unit ball $\bar{B}_{\ell_\infty}$ of $\ell_\infty$ contains an analytic copy of $B_{\ell_\infty}$. In the case of $\mathcal{A}_u(B_X)$ we prove that if there exists a polynomial whose restriction to the open unit ball of $X$ is not weakly continuous at some point, then the fiber over every point of the open unit ball of the bidual contains an analytic copy of $\mathbb{D}$.