Boundaries for algebras of holomorphic functions

Let A u ( B G ) be the Banach algebra of all complex valued functions defined on the closed unit ball B G of a complex Banach space G which are uniformly continuous on B G and holomorphic in the interior of B G , endowed with the sup norm. A characterization of the boundaries for A u ( B G ) is given in case G belongs to a class of Banach spaces that includes the pre-dual of a Lorentz sequence space studied by Gowers in Israel J. Math. 69 (1990) 129–151. The non-existence of the Shilov boundary for A u ( B G ) is also proved.