infinite-dimensional Hilbert space ${{\mathcal{H}}}$.

Part (\[Cuntz\_def\_property\_a\]) of the following definition is originally from [@Cun78] and Part (\[Cuntz\_def\_property\_b\]) is from [@APT11].

\[Cuntz\_def\_property\] Let $A$ be a [C\*-algebra]{}.

1.  \[Cuntz\_def\_property\_a\] For any $a, b \in M_{\infty} (A)_{+}$, we say that $a$ is [*Cuntz subequivalent to $b$ in $A$*]{}, written $a \precsim_{A} b$, if there is a sequence $(c_n)_{n = 1}^{\infty}$ in $M_{\infty} (A)$ such that $${\lim_{{n} \to \infty}} c_n b c_n^* = a.$$ If $a \precsim_{A} b$ and $b \precsim_{A} a$, we say that $a$ and $b$ are [*Cuntz equivalent in $A$*]{} and write $a \sim_{A} b$. This relation is an equivalence relation, and we write $\langle a \rangle_A$ for the equivalence class of $a$. We define ${{\operatorname{W}}}(A) = M_{\infty} (A)_{+} / \sim_A$, together with the commutative semigroup operation $\langle a \rangle_A + \langle b \rangle_A
     = \langle a \oplus b \rangle_A$ and the partial order $\langle a \rangle_A \leq \langle b \rangle_A$ if $a \precsim_{A} b$. We write $0$ for $\langle 0 \rangle_A$. Further, we take ${{\operatorname{Cu}}}(A) = {{\operatorname{W}}}({{\mathcal{K}}}\otimes A)$. We write the classes as $\langle a \rangle_A$ for $a \in ({{\mathcal{K}}}\otimes A)_{+}$.

2.  \[Cuntz\_def\_property\_b\] Let $A$ and $B$ be C\*-algebras and let $\psi \colon A \to B$ be a [homomorphism]{}. 