Uniqueness of Hahn--Banach extensions and inner ideals in real C${}^{\ast }$-algebras and real JB${}^{\ast }$-triples

Abstract

We show that every closed (resp., weak${}^{\ast }$-closed) inner ideal $I$ of a real JB${}^{\ast }$-triple (resp. a real JBW${}^{\ast }$-triple) $E$ is Hahn--Banach smooth (resp., weak${}^{\ast }$-Hahn--Banach smooth). Contrary to what is known for complex JB${}^{\ast }$-triples, being (weak${}^{\ast }$-)Hahn--Banach smooth does not characterise (weak${}^{\ast }$-)closed inner ideals in real JB(W)${}^{\ast }$-triples. We prove here that a closed (resp., weak${}^{\ast }$-closed) subtriple of a real JB${}^{\ast }$-triple (resp., a real JBW${}^{\ast }$-triple) is Hahn-Banach smooth (resp., weak${}^{\ast }$-Hahn-Banach smooth) if, and only if, it is a hereditary subtriple. If we assume that $E$ is a reduced and atomic JBW${}^{\ast }$-triple, every weak${}^{\ast }$-closed subtriple of $E$ which is also weak${}^{\ast }$-Hahn-Banach smooth is an inner ideal.\smallskip In case that $C$ is the realification of a complex Cartan factor or a non-reduced real Cartan factor, we show that every weak${}^{\ast }$-closed subtriple of $C$ which is weak${}^{\ast }$-Hahn-Banach smooth and has rank $\ge 2$ is an inner ideal. The previous conclusions are finally combined to prove the following: Let $I$ be a closed subtriple of a real JB${}^{\ast }$-triple $E$ satisfying the following hypotheses: $(a)$ $I^{\ast }$ is separable. $(b)$ $I$ is weak${}^{\ast }$-Hahn-Banach smooth. $(c)$ The projection of $I^{\ast \ast }$ onto each real or complex Cartan factor summand in the atomic part of $E^{\ast \ast }$ is zero or has rank $\ge 2$. Then $I$ is an inner ideal of $E$.