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The Gelfand-Naimark theorem for \(JB^ *\)-triples. (English) Zbl 0637.46049
The paper contains as an introduction the recent results on JB\({}^*\)- triples. The authors then prove their main result. Let U denote a \(JB^*\)-triple, then U is isometrically isomorphic to a subtriple of an \(\ell^{\infty}\)-direct sum of Cartan factors. From this theorem several corollaries are derived.
(1) U is isometrically isomorphic to a subtriple of \(B(H)\oplus C(S,C^ 6).\)
(2) U is isomorphic to a subtriple of a \(JB^*\)-algebra.
(3) \(\| \{xyz\}\| \leq \| x\| \| y\| \| z\|\) if x,y,z\(\in U.\)
(4) U has a separating family of factor representations.
(5) U is isomorphic to a \(J^*\)-algebra if and only if an analogue of Glennie’s identity is identically zero in U.
Finally the theorem is carried over to the field of bounded symmetric domains.
Reviewer: K.Alvermann

MSC:
46H70 Nonassociative topological algebras
17C65 Jordan structures on Banach spaces and algebras
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