• Maps preserving $A^{\ast} A + AA^{\ast}$ on $C^{\ast}$-algebras

• Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/129/03/0030

• Keywords

$C^{*}$-algebra; $\mathbb{C}$-linear; $\mathbb{C}$-antilinear; homomorphism; linear preserver problem; real rank zero

• Abstract

Let $\mathcal{A}$ be a $C^{*}$-algebra of real-rank zero and $\mathcal{B}$ be a $C^{*}$-algebra with unit $I$. It is shown that the mapping $\Phi: {\mathcal A}\longrightarrow {\mathcal B}$ which preserves arithmetic mean and satisfies$$\Phi(A^{*}A)=\frac{\Phi(A)^{*}\Phi(A)+\Phi(A)\Phi(A)^{*}}{2},$$for all normal elements $A\in \mathcal{A}$, is an $\mathbb R$-linear continuous Jordan $*$-homomorphism provided that $0\in {\rm Ran}\ \Phi$. Also, $\Phi$ is the sum of a linear Jordan $*$-homomorphism and a conjugate-linear Jordan $*$-homomorphism. This result also presents an application of maps which preserve the square absolute value.

• Author Affiliations

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

• Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• Editorial Note on Continuous Article Publication

Posted on July 25, 2019