Let $A$ be a $C^*$-algebra, and let $B$ be a unital $C^*$-algebra such that $A\subset B$ is an essential closed two-sided ideal. For each $b\in B$, define maps
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\begin{align*}
L_b:A&\to A,
\end{align*}
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by $L_b(a)=ba$, and
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\begin{align*}
R_b:A&\to A,
\end{align*}
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by $R_b(a)=ab$. Then there exists a unique unital $*$-homomorphism