$q_{\infty} ( \{ \mathfrak{B}_{n} \} )$ induced by $\{ \phi_{n} \}_{ n = 1}^{\infty}$.

Let $\mathfrak{A}$ be a $C^{*}$-algebra over $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$. For each $n \in \widetilde{{\ensuremath{\mathbb{N}}\xspace}}$, denote the $*$-homomorphism from $\mathfrak{A}$ to $\mathfrak{A}(n)$ by $\pi_{n}$. The quotient map from $\ell^{\infty} ( \{ \mathfrak{A} (n) \} )$ to $q_{\infty} ( \{ \mathfrak{A}( n) \} )$ will be denoted $\rho_{\mathfrak{A}}$ or just $\rho$.

\[l:contsubalg\] Let $\mathfrak A$ be a continuous $C^\ast$-algebra over $\widetilde {\ensuremath{\mathbb{N}}\xspace}$, $\mathfrak B$ be a $C^\ast$-algebra and let $\iota \colon \mathfrak B \to \mathfrak A( \infty ) $ be an injective $*$-homomorphism. Construct the pullback diagram $$\xymatrix{
0 \ar[r] & \mathfrak A({\ensuremath{\mathbb{N}}\xspace}) \ar[r] \ar@{=}[d] & \mathfrak E \ar[r] \ar[d] & \mathfrak B \ar[d]^\iota \ar[r] & 0 \\
0 \ar[r] & \mathfrak A({\ensuremath{\mathbb{N}}\xspace}) \ar[r] & \mathfrak A \ar[r]^-{\pi_{\infty}} & \mathfrak A( \infty ) \ar[r] & 0.
}$$ Then $\mathfrak E$ is a continuous $C^\ast$-algebra over $\widetilde {\ensuremath{\mathbb{N}}\xspace}$ when given the structure $\mathfrak E(U) = \mathfrak E\cap (\mathfrak A(U)\oplus i_\infty(\mathfrak B)(U) )$ for $U\subseteq \widetilde {\ensuremath{\mathbb{N}}\xspace}$ open.

That $\mathfrak E$ is a $C^\ast$-algebra over $\widetilde {\ensuremath{\mathbb{N}}\xspace}$ follows from [@mdrm:ethy Lemma 2.24] by observing that $\pi_{\infty} \colon \mathfrak A \to i_\infty(\mathfrak A(\infty))$ and $\iota \colon i_\infty ( \mathfrak B) \to i_\infty (\mathfrak 