Let $A$ be a $C^*$-algebra, and let $\mathcal{L}(A)$ denote the [Banach space](/page/Banach%20Space) of bounded complex-linear maps $A\to A$ with the operator norm $\|\cdot\|_{\mathcal L(A)}$. A double centralizer of $A$ is a pair $(L,R)$ with $L,R\in\mathcal{L}(A)$ such that, for all $x,y\in A$,
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\begin{align*}
L(xy)=L(x)y.
\end{align*}
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\begin{align*}
R(xy)=xR(y).
\end{align*}
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\begin{align*}
xL(y)=R(x)y.
\end{align*}
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Let $M(A)$ denote the set of all double centralizers, equipped with addition and scalar multiplication componentwise, product
where $L_a(b)=ab$ and $R_a(b)=ba$ for $b\in A$, is an injective $*$-homomorphism. Its image $\iota(A)$ is an essential closed two-sided ideal of $M(A)$.