Abstract
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 maps, that satisfy $$\begin{aligned} \varPhi \left( \frac{AB^*C+CB^*A}{2} \right) =\frac{\varPhi (A)\varPhi (B)^*\varPhi (C)+\varPhi (C)\varPhi (B)^*\varPhi (A)}{2} \end{aligned}$$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.
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