Uniqueness of Hahn--Banach extensions and inner ideals in real C$^*$-algebras and real JB$^*$-triples

Abstract

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