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MAPS PRESERVING JORDAN TRIPLE PRODUCT A<sup>*</sup>B + BA<sup>*</sup> ON *-ALGEBRAS
Taghavi, Ali,Nouri, Mojtaba,Razeghi, Mehran,Darvish, Vahid The Kangwon-Kyungki Mathematical Society 2018 한국수학논문집 Vol.26 No.1
Let $\mathcal{A}$ and $\mathcal{B}$ be two prime ${\ast}$-algebras. Let ${\Phi}:\mathcal{A}{\rightarrow}\mathcal{B}$ be a bijective and satisfies $${\Phi}(A{\bullet}B{\bullet}A)={\Phi}(A){\bullet}{\Phi}(B){\bullet}{\Phi}(A)$$, for all $A,B{\in}{\mathcal{A}}$ where $A{\bullet}B=A^{\ast}B+BA^{\ast}$. Then, ${\Phi}$ is additive. Moreover, if ${\Phi}(I)$ is idempotent then we show that ${\Phi}$ is ${\mathbb{R}}$-linear ${\ast}$-isomorphism.
A NOTE ON NONLINEAR SKEW LIE TRIPLE DERIVATION BETWEEN PRIME ⁎-ALGEBRAS
Taghavi, Ali,Nouri, Mojtaba,Darvish, Vahid The Kangwon-Kyungki Mathematical Society 2018 한국수학논문집 Vol.26 No.3
Recently, Li et al proved that ${\Phi}$ which satisfies the following condition on factor von Neumann algebras $${\Phi}([[A,B]_*,C]_*)=[[{\Phi}(A),B]_*,C]_*+[[A,{\Phi}(B)]_*,C]_*+[[A,B]_*,{\Phi}(C)]_*$$ where $[A,B]_*=AB-BA^*$ for all $A,B{\in}{\mathcal{A}}$, is additive ${\ast}-derivation$. In this short note we show the additivity of ${\Phi}$ which satisfies the above condition on prime ${\ast}-algebras$.
Maps preserving Jordan and $\ast$-Jordan triple product on operator $\ast$-algebras
Vahid Darvish,Mojtaba Nouri,Mehran Razeghi,Ali Taghavi 대한수학회 2019 대한수학회보 Vol.56 No.2
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.
MAPS PRESERVING JORDAN AND ⁎-JORDAN TRIPLE PRODUCT ON OPERATOR ⁎-ALGEBRAS
Darvish, Vahid,Nouri, Mojtaba,Razeghi, Mehran,Taghavi, Ali Korean Mathematical Society 2019 대한수학회보 Vol.56 No.2
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.