Non-linear product $LM^\ast -ML^\ast $ on prime $\ast-$algebras
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In this paper, we explore the additivity of the map $\Omega :\mathrsfso{A}\rightarrow \mathrsfso{A}$ that satisfies $$\Omega\left( [\mathrsfso{L},\mathrsfso{M}]_{*} \right)=[\Omega\left( \mathrsfso{M}\right),\mathrsfso{L}] _{*} + [\mathrsfso{M}, \Omega\left( \mathrsfso{L}\right)]_{*},$$ where $[\mathrsfso{L}, \mathrsfso{M}] _{*}= \mathrsfso{L}\mathrsfso{M}^\ast -\mathrsfso{M} \mathrsfso{L}^\ast$, for all $\mathrsfso{L},\mathrsfso{M} \in\mathcal {\mathrsfso{A} }$, a prime $\ast-$algebra with unit $\mathrsfso{I}$. Additionally we show that if ${\Omega}(\alpha \mathrsfso{I})$ is self-adjoint operator for $ \alpha \in\{1, i\} $, then $\Omega=0$.
Abstract
In this paper, we explore the additivity of the map $\Omega :\mathrsfso{A}\rightarrow \mathrsfso{A}$ that satisfies $$\Omega\left( [\mathrsfso{L},\mathrsfso{M}]_{*} \right)=[\Omega\left( \mathrsfso{M}\right),\mathrsfso{L}] _{*} + [\mathrsfso{M}, \Omega\left( \mathrsfso{L}\right)]_{*},$$ where $[\mathrsfso{L}, \mathrsfso{M}] _{*}= \mathrsfso{L}\mathrsfso{M}^\ast -\mathrsfso{M} \mathrsfso{L}^\ast$, for all $\mathrsfso{L},\mathrsfso{M} \in\mathcal {\mathrsfso{A} }$, a prime $\ast-$algebra with unit $\mathrsfso{I}$. Additionally we show that if ${\Omega}(\alpha \mathrsfso{I})$ is self-adjoint operator for $ \alpha \in\{1, i\} $, then $\Omega=0$.
참고문헌 (11)
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