Let $X$ be a second countable locally compact Hausdorff space, let $C_0(X)$ be the $C^*$algebra of continuous functions vanishing at infinity.
Definition: $C_0(X)$ is called quasi diagonal, if $C_0(X)$ admits a faithful *-representation $\pi:C_0(X)\to B(H)$ and an increasing sequence $(p_n)_n$ of finite rank of projections on $H$ that converges strongly to $id_H$ and such that $$\lim\limits_{n\to\infty}\|p_n\pi(a)-\pi(a)p_n\|=0.$$
The task is to prove that for $C_0(X)$.
Consider $\pi=\bigoplus\limits_{x\in D}ev_x:C_0(X)\to \bigoplus\limits_{x\in D}\mathbb{C}$ given by $\pi(f)=\sum\limits_{x\in D}f(x)$, ($ev_x$ denotes the evaluation at $x$) where $D$ is a countable dense subset of $X$.
Next step is to construct a sequence of finite rank projections $(p_n)\subset \pi(C_0(X))$ with the following properties:
(i) $p_n\le p_{n+1}$ for all $n$, (ii) $\|ap_n-p_na\|\xrightarrow{n\to\infty}0$ for all $a\in \pi(C_0(X))$ and (iii) $\|p_n(x)-x\|\xrightarrow{n\to\infty}0$ for all $x\in \bigoplus\limits_{x\in D}\mathbb{C}$.
My plan is to construct the sequence in the following way: choose $p_1$ as the projection on $\bigoplus\limits_{x\in D}\mathbb{C}$ onto the first coordinate of $\bigoplus\limits_{x\in D}\mathbb{C}$, $p_2$ as the projection on $\bigoplus\limits_{x\in D}\mathbb{C}$ onto the first two coordinates of $\bigoplus\limits_{x\in D}\mathbb{C}$ and so on. To check that $(p_n)\subset \pi(C_0(X))$, it probably works with Urysohn's lemma.
Question: Does this construction of $(p_n)$ work, or do I fail to notice something? If it's wrong, which sequence is a correct one? Or, how would you do it?
It is usually not true that $p_n\in \pi(C_0(X))$ (it may work for carefully chosen $X$, but not in general).
Other than that, your construction works. In fact, $\pi(a)p_n-p_n\pi(a)=0$ for all $n$ and all $a$.