A Bourgain-Pisier construction for general Banach spaces

We prove that every Banach space, not necessarily separable, can be isometrically embedded into a ${\mathcal{L}}_{\mathrm{\infty }}$-space in a way that the corresponding quotient has the Radon-Nikodym and the Schur properties. As a consequence, we obtain ${\mathcal{L}}_{\mathrm{\infty }}$ spaces of arbitrary large densities with the Schur and the Radon-Nikodym properties. This extents the a classical result by J. Bourgain and G. Pisier.