Javier Merì, Universidad de Granada, E-mail address: jmeri@ugr.es

FINITE-DIMENSIONAL BANACH SPACES WITH

NUMERICAL INDEX ZERO

(Joint work with Miguel Martín and Angel Rodríguez-Palacios)

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^{iq_{1}ρ}x_{1} + ⋯ + e^{iq_{n}ρ}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.