I previously asked this question on MathOverflow and received the answer that $c_0$ is an example of a $C^*$-algebra which is not von Neumann but satisfies the property that every self-adjoint element is the limit of a sequence of linear combinations of orthogonal idempotents.
I think I managed to show that $c_0$ is a $C^*$-algebra that is not von Neumann. However, I am having trouble showing that every self-adjoint element is a limit of a sequence of linear combinations of orhogonal idempotents. Can anyone please show me how $c_0$ has this property?
I know that, if $x \in c_0$ is self-adjoint, then $x$ is a real sequence converging to zero. How I can show that $x$ is a limit of sequence of linear combinations of orthogonal idempotents, I am not quite sure?
You don't even need selfadjoint, because all elements of $c_0$ are normal. Every element of $c_0$ is of the form $$\tag{1} x=\sum_{j=1}^\infty x_j\, e_j, $$ where $e_j$ is the characteristic function of $\{j\}$ i.e., $$ e_j=(0,\ldots,0,1,0,0,\ldots). $$ As the series in $(1)$ converges in norm, we have $$ x=\lim_{n\to\infty}\sum_{j=1}^n x_j\, e_j, $$ and each sum is a linear combination of projections.