Mankiewicz’s theorem and the Mazur–Ulam property for $\mathbf {C}^*$-algebras
Volume 250 / 2020
Studia Mathematica 250 (2020), 265-281
MSC: Primary 46B20; Secondary 46B04, 46L05.
DOI: 10.4064/sm180727-14-11
Published online: 6 August 2019
Abstract
We prove that every unital $\mathrm {C}^*$-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 F. J. Fernández-Polo and A. M. Peralta 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.
