) \ar[r]_{\rho_{\mathfrak{A}}} \ar[d]^{\iota_{\mathfrak{A}}} & q_{\infty} ( \{ \mathfrak A( n ) \} ) \ar[r] \ar[d]^{\overline \iota_{ \mathfrak{A} }} & 0 \\
0 \ar[r] & c_{0} (\{ \mathfrak A( n ) \} ) \ar[r] & \mathcal{M} ( c_{0} (  \{ \mathfrak A( n ) \} ) )\ar[r] & \mathcal{Q} ( c_{0}( \{ \mathfrak A( n ) \} )) \ar[r] & 0,
}$$ where $\pi_n \colon \mathfrak A \to \mathfrak A ( n )$ are the canonical epimorphisms, $\iota$ is the canonical inclusion and $\overline \tau_{\mathfrak{A}}$ and $\overline \iota$ are the unique induced $*$-homomorphisms. Let $\tau_{\mathfrak{A}}$ denote the Busby map of the top row. Then $\tau_{\mathfrak{A}} = \overline \iota \circ \overline \tau_{\mathfrak{A}}$. Also, $$\begin{aligned}
\mathfrak{A} \cong \mathfrak{A} (  \infty  ) \oplus_{ \overline{\tau}_{\mathfrak{A}} , \rho_{\mathfrak{A}} } \ell^{\infty} ( \{ \mathfrak{A}(n) \} ) \end{aligned}$$ via the $*$-isomorphism $a \mapsto ( \pi_{\infty}(a), \{ \pi_{n} ( a ) \}_{ n = 1}^{\infty} )$.

That $\mathfrak{A} \cong \mathfrak{A} (  \infty  ) \oplus_{ \overline{\tau}_{\mathfrak{A}} , \rho_{\mathfrak{A}} } \ell^{\infty} ( \{ \mathfrak{A}(n) \} ) $ follows by a diagram chase in the top part of the diagram. Let $\sigma : \mathfrak A \to \mathcal M(\mathfrak A({\ensuremath{\mathbb{N}}\xspace})) \cong \ell^\infty ( \{ \mathcal M(\mathfrak A( n )) \} )$ denote the 