Let $A$ be a $C^*$-algebra and $J$ a closed two-sided ideal of $A$. I want to show that $A/J$ with the quotient norm is a $C^*$-algebra as well, for the $*$-operation $(a+J)^*:= a^*+J$.
In the book I'm reading, the author provides a proof in the case that $A$ has a unit, so I can assume my algebra is non-unital.
Attempt:
Consider the unitalisation $A_I$. We have that $J$ is a norm-closed twosided ideal in $A_I$ and consequently, we see that $A_I/J$ is a $C^*$-algebra. Since $A \subseteq A_I$, we see that
$$A/J \subseteq A_I/J$$
and thus the $C^*$-identity in $A_I/J$ is inherited by $A/J$, which ends the proof.
Is this correct?
One small detail you may want to mention is that the inclusion $A/J\hookrightarrow A_I/J$ is isometric. Other than that, everything seems fine.