Theorem 36.13.3.label Let $A$ be a unital $C^{*}$-algebra and $I \subset A$ be a closed two-sided ideal, then $A/I$ equipped with the quotient norm is a $C^{*}$-algebra.
Proof. Under the quotient structures, $A/I$ is an involutive Banach algebra. It remains to show that $\norm{x + I}_{A/I}^{2} = \norm{x^*x + I}_{A/I}$ for all $x \in A$.
By Lemma 36.13.1, $I$ is a $C^{*}$-algebra. Thus Theorem 36.12.5 implies the existence of an increasing approximate identity $\angles{e_\beta}_{\beta \in B}\subset I$ for $I$.
For each $x \in A$ and $y \in I$,
\begin{align*}\norm{x + I}_{A/I}^{2}&= \lim_{\beta \in B}\norm{x - xe_\beta }_{A}^{2} = \lim_{\beta \in B}\norm{(x - xe_\beta )^*(x - x e_\beta)}_{A} \\&= \lim_{\beta \in B}\norm{(1_A - e_\beta)x^*x(1_A - e_\beta)}_{A} \\&= \lim_{\beta \in B}\norm{(1_A - e_\beta)(x^*x + y)(1_A - e_\beta)}_{A} \le \norm{x^*x + y}_{A}\end{align*}
As the above holds for all $y \in I$, $\norm{x + I}_{A/I}^{2} \le \norm{x^*x + I}_{A/I}$. Thus $A/I$ equipped with the quotient norm is a $C^{*}$-algebra.$\square$
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