A noncommutative Amir-Cambern theorem for von Neumann algebras and nuclear ${C}^∗$-algebras.

Éric Ricard (1, 2) , Jean Roydor (3)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme
3 IMB - Institut de Mathématiques de Bordeaux

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.

Domaines: Analyse fonctionnelle [math.FA]

Éric Ricard, Jean Roydor. A noncommutative Amir-Cambern theorem for von Neumann algebras and nuclear ${C}^∗$-algebras.. J. Funct. Anal. 267 (2014), 2014, pp.J. Funct. Anal. 267 (2014), no. 4, 1121-1136. ⟨hal-01016334⟩