= Spectral values attained by C-star states
{title2=$\lambda\in\sigma(x)\Longrightarrow\exists f\in S(A):f(x)=\lambda$}
On the linear span of $1,x$, define $g(a1+bx)=a+b\lambda$. Affine spectral mapping gives $|a+b\lambda|\leq\|a1+bx\|$, making this well-defined with norm one and $g(1)=1$. A norm-preserving <Hahn-Banach theorem> extension to the whole <C-star algebra> remains unital; the norm characterization of a <positive functional on a C-star algebra> makes it a <state on a C-star algebra> attaining the spectral value.
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