Talk:Stone-Weierstrass theorem
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Stone-Weierstrass doesn't work in the complex case. Consider complex functions on the unit circle in the complex plane. Polynomials form an algebra with a unit and containing functions which separate the points on the circle. However, the function z->complex_conjugate(z) is *not* well approximated by any polynomial. Indeed, conventional "inner product" between this function and any polynomial is 0 (because the complex conjugate of the complex conjugate is again z).
The solution is to consider *-algebras, that is, algebras which are also closed under the operation of complex conjugation. -- Miguel