Analytic structure in fibers
Volume 240 / 2018
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 $\overline {B}_{\ell _{\infty }}$ 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 $B_{X}$ 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 $\mathbb {D}.$
