A noncommutative Amir–Cambern theorem for von Neumann algebras and nuclear C⁎C⁎-algebras

We prove that von Neumann algebras and separable nuclear C⁎C⁎-algebras are stable for the Banach–Mazur cb-distance. A technical step is to show that unital almost completely isometric maps between C⁎C⁎-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⁎C⁎-algebras are close enough for the cb-distance, then they have comparable length.