A fibered description of the vector-valued spectrum

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 \(\mathcal H^\infty(B_Y, X^{**})\). The aim of this article is to describe the fibers defined by this projection, searching for analytic balls and considering Gleason parts.