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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.

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 |