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Geometric characterization of tripotents in real and complex JB\(^{*}\)-triples. (English) Zbl 1058.46033
The authors show that the geometric characterization of tripotent elements (in other words, partial isometries) in \(C^*\)-algebras obtained by C. A. Akemann and N. Weaver [Proc. Am. Math. Soc. 130, No. 10, 3033–3037 (2002; Zbl 1035.46038)] is also valid for \(JB^*\)-triples (every \(C^*\)-algebra is a \(JB^*\)-triple for the triple product defined by \(\{a,b,c\}= 1/2(ab^*c+ cb^*a)\), and the same norm). As a consequence, they obtain an alternative proof of W. Kaup’s Banach-Stone theorem for \(JB^*\)-triples [Math. Z. 183, 503–529 (1983; Zbl 0519.32024)]. The basic geometric tool used in the proofs involves results on the \(M\)-structure in \(JB^*\)-triples and \(JBW^*\)-triples and on the dual \(L\)-structure in their dual and preduals.

MSC:
46K70 Nonassociative topological algebras with an involution
17C65 Jordan structures on Banach spaces and algebras
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