States separate elements of a C-star algebra ($x\ne 0⟹\exists f\in S(A):f(x)\ne 0$)

Write $x=h+ik$ with both terms Hermitian. At least one is nonzero; its norm equals its spectral radius, so it has a nonzero spectral value. Spectral values attained by C-star states gives a state taking that nonzero value on the chosen term. Both state values on $h,k$ are real, so the corresponding complex value on $x$ cannot vanish. This also detects nonzero quasinilpotent elements, which cannot be separated by looking only at their own spectrum.