Abstract
It is shown that every proper weak* closed face of the closed unit ball \({E_1^*}\) in the dual space of a JB*-triple E coincides with set of all elements in the unit sphere of E* attaining their norm at a unique compact tripotent in E**. In particular every proper weak* closed face of the closed unit ball \({E_1^*}\) is weak*-semi-exposed. This result provides an affirmative answer to a conjecture posed over 20 years ago.
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Fernández-Polo, F.J., Peralta, A.M. On the facial structure of the unit ball in the dual space of a JB*-triple. Math. Ann. 348, 1019–1032 (2010). https://doi.org/10.1007/s00208-010-0511-9
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DOI: https://doi.org/10.1007/s00208-010-0511-9