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

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 $(X_n,\|\cdot\|_n)_{n\in\mathbb{N}}$, we define the Bourgain Delbaen $\mathcal{L}^{\infty}$-sum of the sequence $(X_n,\|\cdot\|_n)_{n\in\mathbb{N}}$ which is a Banach space $\mathcal{Z}$ constructed with the Bourgain-Delbaen method. In particular, for every $1\leq p<\infty$, 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 $\mathcal{Z}^n_p=\sum_{i=1}^n\oplus \mathcal{Z}_p$ admits exactly $n+1$, pairwise not isomorphic, complemented subspaces.

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