On the structure of separable $\mathcal{L}_\infty$-spaces

Abstract: Based on a construction method introduced by J. Bourgain and F. Delbaen, we give a general definition of a Bourgain-Delbaen space and prove that every infinite dimensional separable $\mathcal{L}\infty$-space is isomorphic to such a space. Furthermore, we provide an example of a $\mathcal{L}\infty$ and asymptotic $c_0$ space not containing $c_0$.