Bourgain-Delbaen ${\mathcal{L}}^{\infty }$-sums of Banach spaces

Abstract

Motivated by a problem stated by S.A.Argyros and Th. Raikoftsalis, we introduce a new class of Banach spaces. Namely, for a sequence of separable Banach spaces $(X_n,\Vert \cdot {\Vert }_n{)}_{n\in \mathbb{N}}$, we define the Bourgain Delbaen ${\mathcal{L}}^{\infty }$-sum of the sequence $(X_n,\Vert \cdot {\Vert }_n{)}_{n\in \mathbb{N}}$ which is a Banach space $\mathcal{Z}$ constructed with the Bourgain-Delbaen method. In particular, for every $1\le p<\infty$, taking $X_n={\ell }_p$ for every $n\in \mathbb{N}$ the aforementioned space ${\mathcal{Z}}_p$ is strictly quasi prime and admits ${\ell }_p$ as a complemented subspace. We study the operators acting on ${\mathcal{Z}}_p$ and we prove that for every $n\in \mathbb{N}$, the space ${\mathcal{Z}}_p^n={\sum }_{i=1}^n\oplus {\mathcal{Z}}_p$ admits exactly $n+1$, pairwise not isomorphic, complemented subspaces.

29 pages, no figures