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Maps Preserving Product \(A^*B+B^*A\) on \(C^*\)-Algebras

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Abstract

Let \(\mathcal {A}\) and \(\mathcal {B}\) be two unital \(C^*\)-algebras. It is shown that if a surjective map \( \Phi : \mathcal {A} \rightarrow \mathcal {B}\) satisfies:

$$\begin{aligned} \Phi \left( \frac{A^*B+B^*A}{2}\right) =\frac{\Phi (A)^*\Phi (B)+ \Phi (B)^*\Phi (A)}{2} \end{aligned}$$

for every \(A,B \in \mathcal {A}\), and if \( \Phi \) is injective or \( \Phi (-I)=-I \), then \(\Phi \) is the direct sum of two \(*\)-homomorphisms, one of which is \({\mathbb {C}}\)-linear and the other is conjugate \({\mathbb {C}}\)-linear.

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Acknowledgements

The authors would like to thank anonymous referees for thorough and detailed reports with many helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P. O. Box 47416-1468, Babolsar, Iran

    Ali Taghavi & Saeed Gholampoor

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  1. Ali Taghavi
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  2. Saeed Gholampoor
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Correspondence to Ali Taghavi.

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Communicated by Mohammad S. Moslehian.

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Taghavi, A., Gholampoor, S. Maps Preserving Product \(A^*B+B^*A\) on \(C^*\)-Algebras. Bull. Iran. Math. Soc. 48, 757–767 (2022). https://doi.org/10.1007/s41980-021-00544-4

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  • DOI: https://doi.org/10.1007/s41980-021-00544-4

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