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