Uniqueness of Hahn–Banach extensions and inner ideals in real C⁎-algebras and real JB⁎-triples

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. 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 ≥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 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 at least 2. Then I is an inner ideal of E .