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

Proceedings of the Indian Academy of Sciences. Mathematical sciences

ALI TAGHAVI^1¹ 

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

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

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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.

【 授权许可】

CC BY