Suppose that $\mathfrak{J}$ is a closed ideal of a C*-algebra $\mathfrak{A}$. Let $(\pi_u, H_u)$ be the universal representation of $\mathfrak{A}$. Is there a way to use these data to describe the universal representation of $\mathfrak{A} / \mathfrak{J}$?
What would be the best reference for this? Could not find it in Kadison--Ringrose, vol. 2.
Theorem. There exists an invariant subspace $K\subseteq H_u$, such that $\pi _u(a)$ vanishes on $K$ for every $a$ in $\mathfrak J$. Moreover, the restriction of $\pi _u$ to $K$, once factored through $\mathfrak{A}/\mathfrak J$, is equivalent to the universal representation of $\mathfrak{A}/\mathfrak J$.
Proof. For each state $\varphi \in S(\mathfrak A)$, let $(\pi _\varphi ,H_\varphi ,\xi _\varphi )$ be the GNS representation of $\mathfrak A$ associated to $\varphi $, so that $$ \pi _u=\bigoplus_{\varphi \in S(\mathfrak A)}\pi _\varphi . $$
Viewing $S(\mathfrak{A}/\mathfrak J)$ as the subset of $S(\mathfrak A)$ formed by the states on $\mathfrak A$ which vanish on $\mathfrak J$, consider the subspace $$ K:= \bigoplus_{\psi \in S(\mathfrak{A}/\mathfrak J)}H_\psi \subseteq \bigoplus_{\varphi \in S(\mathfrak A)}H_\varphi = H_u. $$
Since each $H_\psi $ is an invariant subspace, so is $K$, and then we may consider the representation $$ \rho :\mathfrak A\to B(K) $$ obtained by restricting $\pi _u$.
For any $\psi $ in $S(\mathfrak{A}/\mathfrak J)$, any $x\in \mathfrak J$, and any $a$ in $\mathfrak A$, notice that $$ \Vert \pi _\psi (x)\pi _\psi (a)\xi _\psi \Vert ^2 = \langle \pi _\psi (xa)\xi _\psi , \pi _\psi (xa)\xi _\psi \rangle = $$$$ \langle \pi _\psi (a^*x^*xa)\xi _\psi , \xi _\psi \rangle = \psi (a^*x^*xa) = 0, $$ from where we see that $\pi _\psi $ vanishes on $\mathfrak J$, and hence so does $\rho $. Therefore $\rho $ factors through a representation of $\mathfrak{A}/\mathfrak J$, which is clearly equivalent to the universal representation of $\mathfrak{A}/\mathfrak J$.
