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Solution of the contractive projection problem. (English) Zbl 0558.46035
Building on tools developed in an earlier paper [Math. Scand. 52, 279-311 (1983)] the authors show that the range of any contractive projection acting on a \(C^*\)-algebra has a natural structure of Jordan triple system. More generally they prove that the category of \(J^*\)-algebras [in the sense of L. A. Harris, Lecture Nores Math. 364, 13-40 (1974; Zbl 0293.46049)] is stable under the action of norm one projections.
The significance of this result is that it shows that JB\({}^*\)-triples in the sense of W. Kaup [Math. Z. 183, 503-529 (1983; Zbl 0519.32024)] occur naturally in the theory of (associative) operator algebras. Inspired by this result, W. Kaup proved the analogous result for \(JB^*\)-triples using holomorphic methods [Math. Scand. 54, 95-100 (1984)]. This result of Kaup, proved earlier in a geometric form by L. L. Stacho [Acta Sci. Math. 44, 99-124 (1982; Zbl 0505.58008)] has already found an important application in the representation theory of Jordan triple systems [cf. the authors: The Gelfand Naimark theorem for \(JB^*\)-triples, preprint].

MSC:
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
17A40 Ternary compositions
47C15 Linear operators in \(C^*\)- or von Neumann algebras
17A65 Radical theory (nonassociative rings and algebras)
46L40 Automorphisms of selfadjoint operator algebras
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