A non-commutative Amir-Cambern theorem for von Neumann algebras and nuclear $C^{\ast }$-algebras

We prove that von Neumann algebras and separable nuclear $C^{\ast }$-algebras are stable for the Banach-Mazur cb-distance. A technical step is to show that unital almost completely isometric maps between $C^{\ast }$-algebras are almost multiplicative and almost selfadjoint. Also as an intermediate result, we compare the Banach-Mazur cb-distance and the Kadison-Kastler distance. Finally, we show that if two $C^{\ast }$-algebras are close enough for the cb-distance, then they have at most the same length.