Let $A$ be a unital $C^*$-algebra, and let $B \subset A$ be a unital $C^*$-subalgebra whose unit is the unit of $A$. Let
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\begin{align*}
P: A \to A
\end{align*}
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be a [linear map](/page/Linear%20Map) such that $P^2=P$, $\operatorname{Range}(P)=B$, and $\|P\|_{\mathcal{L}(A)}=1$. Then, for every $a\in A$ and every $b_1,b_2\in B$,