Approximate units in AF algebras

Question

I am trying to build an approximate unit whose elements have finite spectrum in any AF-algebra. Suppose $A$ is a C*-algebra which is built as an inductive sequence $(A_n,\lambda_n)$ of finite dimensional C*-algebras. If we consider the maps $\phi^n : A_n\rightarrow A$ then we know $A=\overline{\cup \phi^n(A_n)}$. As each $A_n$ is finite dimensional then there are approximate units for $\overline{\phi^n(A)}$ for each $n$. But I am struggling to use these to build an approximate unit on the whole of $A$

Answer 1

If $A$ is unital, you can take $a_n=1_A$ for all $n$.

If $A$ is not unital, take $a_n=\phi^n(1_{A_n})$. Then, given $a\in A$ and $\varepsilon>0$ there exists $n_0$ and $b\in A_{n_0}$ such that $\|a-\phi^n(b)\|<\varepsilon/2$ (initially for $n=n_0$, but then for all $n\geq n_0$, since $\phi^{n_0+k}(b)=\phi^{n}(b)$). Then, for $n\geq n_0$, $$ \|a-a\,a_n\|\leq \|a-\phi^n(b)|+\|\phi^n(b)a_n-a\,a_n\|\leq 2\|a-\phi^n(b)\|<\varepsilon. $$ Thus $\|a-a\,a_n\|\to0$ and $\{a_n\}$ is an approximate unit.