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Additivity of maps preserving Jordan triple products on prime \(C^*\)-algebras. (English) Zbl 1443.47040

Summary: Let \(\mathcal{A}\) and \(\mathcal{B}\) be two unital \(C^*\)-algebras such that \(\mathcal{A}\) contains a non-trivial projection \(P_1\). In this paper, we investigate the additivity of maps \(\varPhi\) from \(\mathcal{A}\) onto \(\mathcal{B}\) that are bijective and satisfy \[ \varPhi\left(\frac{AB^*C+CB^*A}{2}\right)=\frac{\varPhi(A)\varPhi (B)^*\varPhi (C)+\varPhi (C)\varPhi (B)^*\varPhi (A)}{2} \] for every \(A,B,C\in\mathcal{A}\). Moreover, if \(\mathcal{B}\) is a prime \(C^*\)-algebra and \(\varPhi (I)\) is a positive element, then \(\varPhi\) is a \(*\)-isomorphism.

MSC:

47B48 Linear operators on Banach algebras
46L05 General theory of \(C^*\)-algebras
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