Numerical boundaries for some classical Banach spaces

Globevnik gave the definition of boundary for a subspace A⊂Cb(Ω). This is a subset of Ω that is a norming set for A. We introduce the concept of numerical boundary. For a Banach space X, a subset B⊂Π(X) is a numerical boundary for a subspace A⊂Cb(BX,X) if the numerical radius of f is the supremum of the modulus of all the evaluations of f at B, for every f in A. We give examples of numerical boundaries for the complex spaces X=c0, C(K) and d*(w,1), the predual of the Lorentz sequence space d(w,1). In all these cases (if K is infinite) we show that there are closed and disjoint numerical boundaries for the space of the functions from BX to X which are uniformly continuous and holomorphic on the open unit ball and there is no minimal closed numerical boundary. In the case of c0, we characterize the numerical boundaries for that space of holomorphic functions.