Maps preserving Jordan and $\ast$-Jordan triple product on operator $\ast$-algebras
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Let $\mathcal{A}$ and $\mathcal{B}$ be two operator $\ast$-rings such that $\mathcal{A}$ is prime. In this paper, we show that if the map $\Phi:\mathcal{A}\to\mathcal{B}$ is bijective and preserves Jordan or $\ast$-Jordan triple product, then it is additive. Moreover, if $\Phi$ preserves Jordan triple product, we prove the multiplicativity or anti-multiplicativity of $\Phi$. Finally, we show that if $\mathcal{A}$ and $\mathcal{B}$ are two prime operator $\ast$-algebras, $\Psi:\mathcal{A}\to\mathcal{B}$ is bijective and preserves $\ast$-Jordan triple product, then $\Psi$ is a $\mathbb{C}$-linear or conjugate $\mathbb{C}$-linear $\ast$-isomorphism.
Abstract
Let $\mathcal{A}$ and $\mathcal{B}$ be two operator $\ast$-rings such that $\mathcal{A}$ is prime. In this paper, we show that if the map $\Phi:\mathcal{A}\to\mathcal{B}$ is bijective and preserves Jordan or $\ast$-Jordan triple product, then it is additive. Moreover, if $\Phi$ preserves Jordan triple product, we prove the multiplicativity or anti-multiplicativity of $\Phi$. Finally, we show that if $\mathcal{A}$ and $\mathcal{B}$ are two prime operator $\ast$-algebras, $\Psi:\mathcal{A}\to\mathcal{B}$ is bijective and preserves $\ast$-Jordan triple product, then $\Psi$ is a $\mathbb{C}$-linear or conjugate $\mathbb{C}$-linear $\ast$-isomorphism.
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