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J. Korean Math. Soc. 2017; 54(3): 867-876
Online first article December 27, 2016 Printed May 1, 2017
https://doi.org/10.4134/JKMS.j160286
Copyright © The Korean Mathematical Society.
Maps preserving $\eta$-product $A^{*}B+\eta BA^{*}$ on $C^{*}$-algebras
Vahid Darvish, Haji Mohammad Nazari, Hamid Rohi, and Ali Taghavi
University of Mazandaran, University of Mazandaran, University of Mazandaran, University of Mazandaran
Let $\mathcal{A}$ and $\mathcal{B}$ be two $C^{*}$-algebras such that $\mathcal{A}$ is prime. In this paper, we investigate the additivity of maps $\Phi$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective and satisfy $$\Phi(A^{*}B+\eta BA^{*})=\Phi(A)^{*}\Phi(B)+\eta \Phi(B)\Phi(A)^{*}$$ for all $A, B\in \mathcal{A}$ where $\eta$ is a non-zero scalar such that $\eta\neq \pm1$. Moreover, if $\Phi(I)$ is a projection, then $\Phi$ is a $\ast$-isomorphism.
Keywords: maps preserving $\eta$-product, $\ast$-isomorphism, prime $C^{*}$-algebras
MSC numbers: 47B48, 46L10
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