Holomorphic functions with large cluster sets

Abstract

We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and every point of the closed unit ball of the bidual in the infinite dimensional case). We show that this set is strongly c-algebrable for all separable Banach spaces. For specific spaces including ℓ p or duals of Lorentz sequence spaces, we have strongly c-algebrability and spaceability even for the subalgebra of uniformly continous holomorphic functions on the ball.