\mathfrak{A}_{n} \} ) = {\ensuremath{\left\{  \{ a_{n} \}_{ n = 1}^{ \infty }  \: : \:  a_{n} \in \mathfrak{A}_{n} , \ \lim_{ n \to \infty } \| a_{n} \| = 0  \right\}}}$,

$\displaystyle \ell^{\infty} ( \{ \mathfrak{A}_{n} \} ) = {\ensuremath{\left\{  \{ a_{n} \}_{ n = 1}^{\infty }  \: : \:  \text{$a_{n} \in \mathfrak{A}_{n}$ and $\{ a_{n} \}_{ n = 1}^{\infty}$ is bounded} \right\}}}$,

$\displaystyle q_{\infty} ( \{ \mathfrak{A}_{n} \} ) = \ell^{\infty} ( \{ \mathfrak{A}_{n} \} ) / c_{0} ( \{ \mathfrak{A}_{n} \} )$.

If $\{ \mathfrak{B}_{n} \}_{ n = 1}^{\infty}$ is a sequence of $C^{*}$-algebras and if $\{ {  \phi_{n}  \colon  \mathfrak{A}_{n}  \rightarrow  \mathfrak{B}_{n}  } \}_{ n = 1}^{\infty}$ is a sequence of $*$-homomorphisms, then $c_{0} ( \{ \phi_{n} \} )$ will denote the $*$-homomorphism from $c_{0} ( \{ \mathfrak{A}_{n} \} )$ to $c_{0} ( \{ \mathfrak{B}_{n} \} )$ induced by $\{ \phi_{n} \}_{ n = 1}^{\infty}$, $\ell^{\infty} ( \{ \phi_{n} \} )$ will denote the $*$-homomorphism from $\ell^{\infty} ( \{ \mathfrak{A}_{n} \} )$ to $\ell^{\infty} ( \{ \mathfrak{B}_{n} \} )$ induced by $\{ \phi_{n} \}_{ n = 1}^{\infty}$, and $q_{\infty} ( \{ \phi_{n} \} )$ will denote the $*$-homomorphism from $q_{\infty} ( \{ \mathfrak{A}_{n} \} )$ to 