The algebra of bounded-type holomorphic functions on the ball

Abstract

We study the spectrum Mb(U) of the algebra of bounded-type holomorphic functions on a complete Reinhardt domain in a symmetrically regular Banach space E as an analytic manifold over the bidual of the space. In the case that U is the unit ball of ℓp, 1 < p < ∞, we prove that each connected component of Mb(Bℓp) naturally identifies with a ball of a certain radius. We also provide estimates for this radius and in many natural cases we have the precise value. As a consequence, we obtain that for connected components different from that of evaluations, these radii are strictly smaller than one, and can be arbitrarily small. We also show that for other Banach sequence spaces, connected components do not necessarily identify with balls.