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We prove that von Neumann algebras and separable nuclear $C^*$-algebras arestable for the Banach-Mazur cb-distance. A technical step is to show thatunital almost completely isometric maps between $C^*$-algebras are almostmultiplicative and almost selfadjoint. Also as an intermediate result, wecompare 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, thenthey have at most the same length.

A non-commutative Amir-Cambern theorem for von Neumann algebras and nuclear $C^*$-algebras