We use the same letter for the induced maps $M_n (A) \to M_n (B)$ for $n \in {{\mathbb{Z}}_{> 0}}$ and ${M_{\infty}}(A) \to {M_{\infty}}(B)$. We define ${{\operatorname{W}}}(\psi) \colon {{\operatorname{W}}}(A) \to {{\operatorname{W}}}(B)$ and ${{\operatorname{Cu}}}(\psi) \colon {{\operatorname{Cu}}}(A) \to {{\operatorname{Cu}}}(B)$ by $\langle a \rangle_A \mapsto \langle \psi (a) \rangle_B$ for $a \in M_{\infty} (A)_{+}$ or $a \in ({{\mathcal{K}}}\otimes A)_{+}$ as appropriate.

The usual notation for Cuntz subequivalence is $a \precsim b$. Since we need to use Cuntz subequivalence with respect to different C\*-algebras, we include $A$ in the notation. The following lemma is taken from [@KR00; @Ph14; @Ror92].

\[PhiB.Lem\_18\_4\] Let $A$ be a [C\*-algebra]{}.

1.  \[PhiB.Lem\_18\_4\_10\] Let $a, b \in A_{+}$ and let $\eta > 0$. If $\| a - b \| < \eta$, then:

    1.  \[PhiB.Lem\_18\_4\_10.a\] $(a - \eta)_{+} \precsim_A b$.

    2.  \[Item\_9420\_LgSb\_1\_6\] For any ${\lambda}> 0$, we have $(a - {\lambda}- \eta)_{+} \precsim_A (b - {\lambda})_{+}$.

2.  \[PhiB.Lem\_18\_4\_8\] Let $a \in A_{+}$ and let $\eta_1, \eta_2 > 0$. Then $$\big( ( a - \eta_1)_{+} - \eta_2 \big)_{+}
     = \big( a - ( \eta_1 + \eta_2 ) \big)_{+}.$$

3.  \[PhiB.Lem\_18\_4\_11\] Let $a, b \in A_{+}$. Then [the following are equivalent]{}:

    1.  \[PhiB.Lem\_18\_4\_11.a\] $a \precsim_A b$.

    2.  \[PhiB.Lem\_18\_4\_11.b\] $(a - \eta)_{+} \precsim_A b$ for 