Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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MAPS PRESERVING JORDAN AND ⁎-JORDAN TRIPLE PRODUCT ON OPERATOR ⁎-ALGEBRAS

  • Darvish, Vahid (Department of Mathematics Faculty of Mathematical Sciences University of Mazandaran) ;
  • Nouri, Mojtaba (Department of Mathematics Faculty of Mathematical Sciences University of Mazandaran) ;
  • Razeghi, Mehran (Department of Mathematics Faculty of Mathematical Sciences University of Mazandaran) ;
  • Taghavi, Ali (Department of Mathematics Faculty of Mathematical Sciences University of Mazandaran)
  • Received : 2018.04.06
  • Accepted : 2019.03.04
  • Published : 2019.03.31
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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 ${\Phi}:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ 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, ${\Psi}:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ 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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Acknowledgement

Supported by : University of Mazandaran

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