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

## TL;DR

This paper proves that surjective isometries from the unit sphere of rank-2 Cartan factors extend to linear isometries, solving an open problem and confirming the Mazur–Ulam property for JBW*-triples.

## Contribution

It establishes the extension property for isometries of rank-2 Cartan factors and JBW*-triples, resolving an open problem in the field.

## Key findings

- Surjective isometries extend to linear isometries for rank-2 Cartan factors.
- The result applies to spin factors as well.
- Confirms the Mazur–Ulam property for JBW*-triples.

## 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$^*$-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 $Y$.