Numerical boundaries for some classical Banach spaces

Globevnik gave the definition of boundary for a subspace A ⊂ C b ( 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 ⊂ C b ( B X , 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 = c 0 , 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 B X 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 c 0 , we characterize the numerical boundaries for that space of holomorphic functions.