Show that the span of elements of the form $\displaystyle a^*b$ where both $\displaystyle a,b\in A$ which is a C*-albegra is dense in $\displaystyle A$
(I'm missing something simple...)
If the algebra is unital, with identity element e, then this result is obvious, because any element $\displaystyle a\in A$ is equal to $\displaystyle e^*a$.
Even in the nonunital case, a C*-algebra always has an approximate unit consisting of positive elements in the unit ball. Given $\displaystyle a\in A$ and $\displaystyle \varepsilon>0$, choose an element $\displaystyle e_\iota$ in the approximate unit such that $\displaystyle \|e_\iota a-a\|<\varepsilon$. That shows that elements of the form $\displaystyle b^*a$ are dense in A (without even having to take the linear span of them).