Characterizing Linear Mappings Through Unital Algebra

Atteya, Mehsin Jabel

Indonesian Journal of Mathematics and Applications
Vol. 3 No. 1 (2025): Indonesian Journal of Mathematics and Applications

In this paper, we characterize the two linear mappings $\sigma$ and $\tau$ satisfying the identity, $x \circ y^{\star}=0$ yields $\sigma(x)\circ y^{\star}+x\circ \tau(y)^{\star}=0$ for all $x, y \in A$, where $A$ is an $\star$-algebra over a real or complex field $K$ from a unital algebra into its unital $\star$-bimodule. Moreover, we push a complete description of linear mapping that $\sigma$ is a linear mapping from $A$ into $M$ satisfying $X, Y \in A$, $X \circ Y=0$ yields $\sigma(X)\circ Y-X\circ \sigma(Y)=0$ and each element of $A$ has a weak inverse.