Edwards, C. Martin; Fernández-Polo, Francisco J.; Hoskin, Christopher S.; Peralta, Antonio M. On the facial structure of the unit ball in a JB*-triple. (English) Zbl 1204.46039 J. Reine Angew. Math. 641, 123-144 (2010). 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. Reviewer: Antonio Jesus Calderón Martín (Puerto Real) Cited in 2 ReviewsCited in 16 Documents MSC: 46L70 Nonassociative selfadjoint operator algebras 46A55 Convex sets in topological linear spaces; Choquet theory Keywords:JB*-triple; facial structure Citations:Zbl 0759.46050 PDFBibTeX XMLCite \textit{C. M. Edwards} et al., J. Reine Angew. Math. 641, 123--144 (2010; Zbl 1204.46039) Full Text: DOI References: [1] DOI: 10.1112/plms/s3-64.2.418 · Zbl 0759.46050 · doi:10.1112/plms/s3-64.2.418 [2] DOI: 10.1093/qmath/41.3.255 · Zbl 0728.46046 · doi:10.1093/qmath/41.3.255 [3] Barton T. J., Math. Scand. 59 pp 177– (1986) [4] DOI: 10.2307/2047221 · Zbl 0661.46045 · doi:10.2307/2047221 [5] Bunce L. J., Math. Scand. 86 pp 17– (2000) [6] DOI: 10.1093/qmath/hah059 · Zbl 1123.46053 · doi:10.1093/qmath/hah059 [7] Dineen S., Math. Scand. 59 pp 131– (1986) [8] Edwards C. M., Bull. Sci. Math. 104 pp 393– (1980) [9] DOI: 10.1112/jlms/s2-38.2.317 · Zbl 0621.46043 · doi:10.1112/jlms/s2-38.2.317 [10] DOI: 10.1017/S0305004100074740 · Zbl 0853.46070 · doi:10.1017/S0305004100074740 [11] DOI: 10.1112/S0024610706022897 · Zbl 1108.46049 · doi:10.1112/S0024610706022897 [12] Friedman Y., Math. 356 pp 67– (1985) [13] DOI: 10.1215/S0012-7094-86-05308-1 · Zbl 0637.46049 · doi:10.1215/S0012-7094-86-05308-1 [14] Horn G., Math. Scand. 61 pp 117– (1987) [15] DOI: 10.1007/BF01173928 · Zbl 0519.32024 · doi:10.1007/BF01173928 [16] Kaup W., Math. Scand. 54 pp 95– (1984) [17] DOI: 10.1112/plms/83.3.605 · Zbl 1037.46058 · doi:10.1112/plms/83.3.605 [18] Rodríguez A., Quart. J. Math. Oxford 42 pp 99– (1989) [19] DOI: 10.1307/mmj/1029001946 · Zbl 0384.46040 · doi:10.1307/mmj/1029001946 [20] DOI: 10.1017/S0305004100055092 · Zbl 0392.46038 · doi:10.1017/S0305004100055092 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.