A fibered description of the vector-valued spectrum

Abstract

For Banach spaces $X$ and $Y$ we study the vector-valued spectrum ${\mathcal{M}}_{\infty }(B_X,B_Y)$, that is the set of non null algebra homomorphisms from ${\mathcal{H}}^{\infty }(B_X)$ to ${\mathcal{H}}^{\infty }(B_Y)$, which is naturally projected onto the closed unit ball of ${\mathcal{H}}^{\infty }(B_Y,X^{\ast \ast })$. The aim of this article is to describe the fibers defined by this projection, searching for analytic balls and considering Gleason parts.