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Let $A$ be a unital $C^*$-algebra, and let $A \otimes \mathcal{K}$ be the minimal unitization of $A$ (i.e. $A \otimes \mathcal{K}$ is the unitization of $A$ with minimal sub-unital homomorphisms as unitaries). Then we have a canonical surjective $^*$-homomorphism $\tau : A \rightarrow A \otimes \mathcal{K}$ given by the functional calculus. Then

\[main\] Let $A$ be a simple unital AF-algebra such that $K_1(A) = 0$. Then there is a finitely generated subgroup $G \subset A_+$, such that $\tau(G) \subset G \otimes \mathcal{K}$ and $A \cong A \otimes \mathcal{K} / (G \otimes \mathcal{K})$.

For $A = C(X)$ the corollary is Theorem 1.3 in [@CKSS]. For a unital $C^*$-algebra, the corollary is Corollary 1.4 in [@CKSS] and Corollary 2.3 in [@GJS]. Theorem \[main\] is an approximate version of these corollaries.

We follow the proof of [@CKSS Theorem 1.3] in order to adapt it to our situation. First let us find the required $G$. Let $F \subset A_+$ be the set of functionals that have zero integral. Then we define $G$ to be the group generated by all elements of $F$. Clearly $G \subset A_+$ is countable, so it remains to check that $G \subset F$. Fix a sequence of functionals $(g_n) \subset F$. By Urysohn’s lemma, we may find functionals $f_n \in F$ such
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