This is an announcement for the paper “On the extension of isometries between the unit spheres of a JBW$^*$-triple and a Banach space” by Julio Becerra-Guerrerohttps://arxiv.org/search/math?searchtype=author&query=Becerra-Guerrero%2C+J, María Cueto-Avellanedahttps://arxiv.org/search/math?searchtype=author&query=Cueto-Avellaneda%2C+M, Francisco J. Fernández-Polohttps://arxiv.org/search/math?searchtype=author&query=Fern%C3%A1ndez-Polo%2C+F+J, Antonio M. Peraltahttps://arxiv.org/search/math?searchtype=author&query=Peralta%2C+A+M.

Abstract: We prove that every JBW$^*$-triple $M$ with rank one or rank bigger than or equal to three satisfies the Mazur--Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$. We also show that the same conclusion holds if $M$ is not a JBW$^*$-triple factor, or more generally, if the atomic part of $M^{**}$ is not a rank two Cartan factor.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.01460