= Spectrum equals character values in a commutative Banach algebra
{title2=$\sigma_A(a)=\{\phi(a):\phi\in\Phi_A\}$}
For a commutative complex unital <Banach algebra>, a <character of an algebra> annihilates $a-\phi(a)1$, so this element cannot be invertible. Conversely a noninvertible $a-\lambda1$ generates a proper <ideal>, which lies in a <maximal ideal>. The corresponding character then has value $\lambda$ at $a$. Thus the <Gelfand transform> determines every element's spectrum even when the algebra is not semisimple.
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