Mankiewicz's theorem and the Mazur--Ulam property for C*-algebras

Abstract

We prove that every unital C*-algebra $A$ has the Mazur--Ulam property. Namely, every surjective isometry from the unit sphere $S_A$ of $A$ onto the unit sphere $S_Y$ of another normed space $Y$ extends to a real linear map. This extends the result of A. M. Peralta and F. J. Fernandez-Polo who have proved the same under the additional assumption that both $A$ and $Y$ are von Neumann algebras. In the course of the proof, we strengthen Mankiewicz's theorem and prove that every surjective isometry from a closed unit ball with enough extreme points onto an arbitrary convex subset of a normed space is necessarily affine.

arXiv comment15 pages. A few more details added

References in corpus (1)

• Tingley's problem through the facial structure of operator algebras

Cited by in corpus (6)

• Tingley's problem through the facial structure of operator algebras
• A reflection on Tingley's problem and some applications
• Extension of isometries from the unit sphere of a rank-2 Cartan factor
• Isometries between projection lattices of von Neumann algebras
• Metric characterisation of unitaries in JB${}^{\ast }$-algebras
• The Mazur--Ulam property in ${\ell }_{\infty }$-sum and $c_0$-sum of strictly convex Banach spaces