Some geometric properties of disk algebras

In this paper we study some geometrical properties of certain classes of uniform algebras, in particular the ball algebra Au(BX) of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space X. We prove that Au(BX) has k-numerical index 1 for every k, the lushness and also the AHSP. Moreover, the disk algebra A(D), and more in general any uniform algebra whose Choquet boundary has no isolated points, is proved to have the polynomial Daugavet property. Most of those properties are extended to the vector valued version AX of a uniform algebra A.