  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.
