Let $A$ be a $C^*$-algebra, and let $I\subset A$. Then $I$ is a closed two-sided ideal of $A$ if and only if there exist a complex [Hilbert space](/page/Hilbert%20Space) $H$ and a $*$-representation $\pi:A\to B(H)$ such that
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\begin{align*}
I=\ker \pi.
\end{align*}
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Here a $*$-representation means a $*$-homomorphism into $B(H)$, not necessarily nonzero or nondegenerate.