Universal representations of quotient C*-algebras 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.
 A: 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 $\mathfrak{A}/\mathfrak J$.
