Let $A_0$ be the set of all bounded operators over $\ell^2(\mathbb{N})$ which can be represented by (infinite) matrices satisfying $$ \exists\ r \in \mathbb{N} \ \text{ s.t. } a_{i,j}= 0 \ \text{ whenever } |i- j|> r $$
Setting $A= \overline{A_0}$ we can check that $A$ is a C*-algebra.
Consider the closed subspace ( even the C*-subalgebra ) $C= \{a\in B(\ell^2(\mathbb{N})) : a_{ij} = 0 \ \text{for } i\neq j\}$
and for $a= (a_{ij})_{i,j\in \mathbb{N}}\in A$ , define
$$E(a) :=
\begin{bmatrix}
a_{11} & 0 & 0 & \cdots \\
0 & a_{22} & 0 & \cdots \\
0 & 0 & a_{33} & \cdots \\
\vdots & \vdots & & \ddots \\
\end{bmatrix} \in C . $$
$E$ is actually a continuous projection $E^2= E:A\to C$ .
We can check that for all $a \in A$ and all unitaries $u\in C:\ E(u^*au) = E(a)$ .
For a fixed $a\in A$ let $S= \big\{s=\frac{1}{k}\sum_{i=1}^ku_i^*au_i\ \big|\ k\in \mathbb{N},\ u_1,\dots,u_k\in C \text{ are unitaries}\big\}$
the set of all finite averages of that form.
$\textbf{Show that }$ for all $\epsilon > 0$ there exists $s\in S$ such that $\big\|s- E(s)\big\|< \epsilon $
or equivalently that $\big\|s- E(a)\big\|< \epsilon \ $ from the earlier remark $E(u^*au) = E(a)$
I am just looking for a hint if possible.
Writing down explicitly the terms doesn't help, so I am supposing we should use some sort of property of the projection $E$ ( but $A$ is not a Hilbert space itself ) and/or structure of unitaries
( the conjugacy action $a\mapsto u^*au$ being a *-automorphism $A\to A$ ).
Nice question!
Hint: what you want is to get all non-diagonal entries of $S$ to be small. More concretely, write the expression for $s_{kj}$ and think $n^{\rm th}$-roots of unity for your unitaries. You should be able to get $$ s-E(s)=\frac1n\,(a-E(a)). $$