= States separate elements of a C-star algebra
{title2=$x\ne0\Longrightarrow\exists f\in S(A):f(x)\ne0$}
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.
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