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

2018/04/27 by Michiya Mori, Mori, Michiya, Narutaka Ozawa +1
Mathematics · #46B04 #46B20 #46L05 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1804.10674

openalex publication_date 2018/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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