Finite-dimensional Banach spaces with numerical index zero

Abstract

We prove that a finite-dimensional Banach space X has numerical index 0 if and only if it is the direct sum of a real space X 0 and nonzero complex spaces X 1 ,...,X n in such a way that the equality ∥x 0 + e iq1ρ x 1 +...+e iqnρ x n ∥ = ∥x 0 +...+ x n ∥ holds for suitable positive integers q 1 ,...,q n , and every ρ ∈ R and every x j ∈ X j (j = 0, 1,...,n). If the dimension of X is two, then the above result gives X = C, whereas dim(X) = 3 implies that X is an absolute sum of R and C. We also give an example showing that, in general, the number of complex spaces cannot be reduced to one.