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Contractive projections on operator triple systems. (English) Zbl 0547.46048

A \(J^*\)-algebra is a norm closed complex linear subspace of \({\mathcal L}(H,K)\), the bounded linear operators from a Hilbert space H to a Hilbert space K, which is closed under the operation \(a\to aa^*a\). The examples of \(J^*\)-algebras are J\(C^*\)-algebras, their abstract generalizations - J\(B^*\)-algebras (if we exclude the exceptional Jordan algebra \((M^ 8_ 3)^ C)\) ternary rings of operators, \(C^*\)-ternary rings etc. The purpose of the authors is to show that the category of \(C^*\)-triple systems (which are an abstract version of \(J^*\)- algebras) is stable under the action of norm one projections. The main result of the paper is
Theorem 3. Let M be a \(J^*\)-algebra and let \(P:M\to M\) be a linear projection of norm one: \(P^ 2=P, \| P\| =1\). Suppose the range P(M) of P is finite dimensional. Then P(M) is linearly isometric to a \(J^*\)-algebra and therefore a \(C^*\)-triple system.
There are examples showing that Theorem 3 is the best possible in the sense that, in the absence of special assumptions on P, P(M) is not in general a ternary ring of operators or a \(JC^*\)-algebra even M is a \(C^*\)-algebra.
Reviewer: Sh.A.Ayupov

MSC:

46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
17C65 Jordan structures on Banach spaces and algebras
17C50 Jordan structures associated with other structures
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
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