Numerical index of absolute sums of Banach spaces

TL;DR

This paper investigates the numerical index of absolute sums of Banach spaces, establishing conditions for its bounds, providing examples, and analyzing its behavior in various Banach space constructions.

Contribution

It introduces new conditions for the numerical index of sums, including E-sums with RNP, and explores its behavior in spaces with dense unions of subspaces.

Findings

Numerical index of sums is bounded by the infimum of summands' indices.

Equality holds for c0, l1, l∞ sums and certain E-sums with RNP.

Numerical indices of all infinite-dimensional Lp spaces are equal.

Abstract

We study the numerical index of absolute sums of Banach spaces, giving general conditions which imply that the numerical index of the sum is less or equal than the infimum of the numerical indices of the summands and we provide some examples where the equality holds covering the already known case of $c_0$-, $\ell_1$- and $\ell_\infty$-sums and giving as a new result the case of $E$-sums where $E$ has the RNP and $n(E)=1$ (in particular for finite-dimensional $E$ with $n(E)=1$). We also show that the numerical index of a Banach space $Z$ which contains a dense increasing union of one-complemented subspaces is greater or equal than the limit superior of the numerical indices of those subspaces. Using these results, we give a detailed short proof of the already known fact that the numerical indices of all infinite-dimensional $L_p(\mu)$-spaces coincide.