 Star polynomial in one normal element

Since a normal C-star element commutes with its adjoint, words in those two generators can be rearranged into the displayed finite sum. Including the constant term generates the unital star algebra; omitting it generates the nonunital star algebra. Under continuous functional calculus, such a sum is the ordinary function ${\sum }_{j,k}{c}_{jk}{z}^j{\overline{z}}^k$ on the spectrum. The Stone-Weierstrass theorem makes these functions uniformly dense.