Shilov boundary for holomorphic functions on some classical Banach spaces

Abstract

Let A1(BX) be the Banach space of all bounded and continuous functions on the closed unit ball BX of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let Au(BX) be the subspace of A1(BX) of those functions which are uniformly continuous on BX. A subset B ⊂ BX is a boundary for A1(BX) if k f k = supx2B |f(x)| for every f ∈ A1(BX). We prove that for X = d(w,1) (the Lorentz sequence space) and X = C1(H), the trace class operators, there is a minimal closed boundary for A1(BX). On the other hand, for X = S, the Schreier space, and X = K(lp, lq) (1 ≤ p ≤ q < ∞), there is no minimal closed boundary for the corresponding spaces of holomorphic functions.