Extension of isometries from the unit sphere of a rank-2 Cartan factor

Abstract

We prove that every surjective isometry from the unit sphere of a rank-2 Cartan factor $C$ onto the unit sphere of a real Banach space $Y$, admits an extension to a surjective real linear isometry from $C$ onto $Y$. The conclusion also covers the case in which $C$ is a spin factor. This result closes an open problem and, combined with the conclusion in a previous paper, allows us to establish that every JBW${}^{\ast }$-triple $M$ satisfies the Mazur--Ulam property, that is, every surjective isometry from its unit sphere onto the unit sphere of a arbitrary real Banach space $Y$ admits an extension to a surjective real linear isometry from $M$ onto $Y$.

References in corpus (5)

• Tingley's problem through the facial structure of operator algebras
• A reflection on Tingley's problem and some applications
• Mankiewicz's theorem and the Mazur--Ulam property for C*-algebras
• The Mazur-Ulam property for commutative von Neumann algebras
• Extending surjective isometries defined on the unit sphere of ${\ell }_{\infty }(\Gamma )$

Cited by in corpus (4)

• Finite tripotents and finite JBW${}^{\ast }$-triples
• The Daugavet equation for polynomials on C${}^{\ast }$-algebras and JB${}^{\ast }$-triples
• A property in vector-valued function spaces
• Tingley's problem on uniform algebras