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

TL;DR

This paper establishes stability results for von Neumann algebras and nuclear $C^*$-algebras under the Banach-Mazur cb-distance, introducing new technical tools and comparisons with the Kadison-Kastler distance.

Contribution

It proves stability of these algebras for the cb-distance and shows that almost isometric maps are nearly multiplicative and selfadjoint, advancing non-commutative Banach space theory.

Findings

Von Neumann algebras are stable under the cb-distance.

Nuclear $C^*$-algebras are stable under the cb-distance.

Close $C^*$-algebras have the same length.

Abstract

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.