Lemma 36.13.1.label Let $A$ be a unital $C^{*}$-algebra and $I \subset A$ be a closed two-sided ideal, then for each $x \in I$, $x^{*} \in I$ as well.
Proof. By Theorem 36.12.5, there exists an increasing net $\angles{e_\beta}_{\beta \in B}\subset I \cap \ol{B_A(0, 1)}$ such that $e_{\beta} x \to x$ and $xe_{\beta} \to x$ for all $x \in I$. As $A$ is a $C^{*}$-algebra,
\[\norm{e_\beta x - x}_{A} = \norm{x^*e_\beta - x^*}_{A} \to 0\]
Given that $I$ is two-sided, $x^{*}e_{\beta} \in I$ for all $\beta \in B$. Since $I$ is closed, the above implies that $x^{*} \in I$ as well.$\square$
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