A characterization of inner ideals in 𝐽𝐵*-triples

Edwards, C. M.; Rüttimann, G. T.

doi:10.1090/S0002-9939-1992-1102856-1

A characterization of inner ideals in $\textrm {JB}^ *$-triples
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Proc. Amer. Math. Soc. 116 (1992), 1049-1057 Request permission

Abstract:

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

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