A characterization of inner ideals in 𝐽𝐵*-triples

Abstract

It is shown that a norm-closed subtriple B B of a J B ∗ J{B^ * } -triple A A is an inner ideal if and only if every bounded linear functional on B B has a unique norm-preserving extension to a bounded linear functional on A A . It follows that the norm-closed subtriples B B of a C ∗ {C^ * } -algebra A A that enjoy this unique extension property are precisely those of the form e A ∗ ∗ f ∩ A e{A^{ * * }}f \cap A where ( e , f ) (e,f) is a pair of centrally equivalent open projections in the W ∗ {W^ * } -envelope A ∗ ∗ {A^{ * * }} of A A .