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