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