= Spectral values attained by C-star states
{title2=$\lambda\in\sigma(x)\Longrightarrow\exists f\in S(A):f(x)=\lambda$}
Let $A$ be a nonzero \b[unital] <C-star algebra>, $x\in A$, and $\lambda\in\sigma_A(x)$. On the <linear span> of $1,x$, define $g(a1+bx)=a+b\lambda$. The <spectral mapping theorem> for affine <polynomials> gives $|a+b\lambda|\leq\|a1+bx\|$, making $g$ well-defined with <norm> one and $g(1)=1$. A norm-preserving <Hahn-Banach theorem> extension to $A$ remains unital; the norm characterization of a <positive functional on a C-star algebra> makes it a <state on a C-star algebra> attaining $\lambda$.
The unit assumption is necessary for this formulation. The element $x(t)=e^{-t^2}$ of the nonunital algebra $C_0(\mathbb R)$ has zero in its unitization <spectrum>, but every state integrates against a probability measure on $\mathbb R$ and takes a strictly positive value on $x$. The preceding extension proof can be performed in the <C-star unitization>; a state obtained there need not restrict to a <norm>-one functional on the original algebra.
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