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On the facial structure of the unit ball in a JB*-triple. (English) Zbl 1204.46039

The paper under review studies the facial structure of the closed unit ball \(A_1\) in a \(JB^*\)-triple \(A\), by showing that every nonempty norm-closed face \(F\) of \(A_1\) is norm-semi-exposed. Since the class of \(JB^*\)-triples extends the one of \(C^*\)-triples, this result extends the one for \(C^*\)-algebras in [C.A.Akemann and G.K.Pedersen, Proc.Lond.Math.Soc., III.Ser.64, No.2, 418–448 (1992; Zbl 0759.46050)].
More precisely, the authors find a (unique) compact tripotent \(u \in A^{**}\) such that \(F\) agrees with the norm-semi-exposed face of \(A_1\) corresponding to \(u\). To prove this main theorem, several previous results concerning, in particular, functional calculus for \(JB^*\)-triples and the study of the properties of certain norm-closed inner ideals of \(A\) are stated. From here, the authors combine all of these preliminary results in a nice way to get the proof.

MSC:

46L70 Nonassociative selfadjoint operator algebras
46A55 Convex sets in topological linear spaces; Choquet theory

Citations:

Zbl 0759.46050
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References:

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