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Orthogonality preservers in C\(^*\)-algebras, JB\(^*\)-algebras and JB\(^*\)-triples. (English) Zbl 1156.46045
Let \(A\) be a \(C^*\)-algebra or a \(JB^*\)-algebra, and let \(E\) be a \(JB^*\)-triple. The authors give a description of those bounded linear operators \(T: A\rightarrow E\) with the property of preserving orthogonality, which means that \(\{T(a),T(b),y\}=0\) for each \(y\in E\) whenever \(a,b\in A\) are such that \(\{a,b,x\}=0\) for each \(x\in A\) (where \(\{\cdot,\cdot,\cdot\}\) stands for the triple product in both \(A\) and \(E\)). It is shown that if \(T''(\mathbf{1})\) is a von Neumann regular element in the bidual \(E''\) of \(E\), then \(T(A)\) is contained in the Peirce subspace \(E_2''(r(T''(\mathbf{1})))\) of the range tripotent \(r(T''(\mathbf{1}))\) of \(T''(\mathbf{1})\) and there exists a triple homomorphism \(S\colon A\rightarrow E_2''(r(T''(\mathbf{1})))\) such that \(T(x)=\{T''(\mathbf{1}),r(T''(\mathbf{1})),S(x)\}\) for each \(x\in A\). As a consequence, it is shown that a bounded linear operator \(T: A\rightarrow B\) between two \(C^*\)-algebras \(A\) and \(B\) preserves orthogonality in the usual sense (\(a,b\in A, \;ab^*=b^*a=0 \;\Rightarrow \;T(a)T(b)^*=T(b)^*T(a)=0\)) if and only if there exists a triple homomorphism \(S\colon A\rightarrow B''\) such that \(T''(\mathbf{1})^*S(x)=S(x^*)^*T''(\mathbf{1})\), \(T''(\mathbf{1})S(x^*)^*=S(x)T''(\mathbf{1})^*\), and \(T(x)=T''(\mathbf{1})r(T''(\mathbf{1}))^*S(x)\) for each \(x\in A\).

MSC:
46L70 Nonassociative selfadjoint operator algebras
46L05 General theory of \(C^*\)-algebras
46H70 Nonassociative topological algebras
46K70 Nonassociative topological algebras with an involution
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