Shilov boundary for holomorphic functions on some classical Banach spaces

Let A infinity(B-x) be the Banach space of all bounded and continuous functions on the closed unit ball B-x of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let A(u)(B-x) be the subspace of A infinity(B-x) of those functions which are uniformly continuous on B-x. A subset B subset of B-x is a boundary for A infinity(B-x) if parallel to f parallel to = SUPx is an element of B vertical bar f(x)vertical bar for every f is an element of A infinity(B-x). We prove that for X = d(w, 1) (the Lorentz sequence space) and X = C-1(H), the trace class operators, there is a minimal closed boundary for A infinity(B-x). On the other hand, for X = S, the Schreier space, and X = K(l(p), l(q)) (1 <= p < q < infinity), there is no minimal closed boundary for the corresponding spaces of holomorphic functions.