= Gelfand representation theorem
{c}
{title2=$a\mapsto\widehat a\in C(\Delta(A))$}
For a commutative complex unital <Banach algebra>, its <Gelfand transform> is a continuous unital algebra homomorphism into the continuous functions on its compact <character space>. The values of the transform of $a$ are exactly $\sigma_A(a)$, so $\|\widehat a\|_\infty=r(a)$. Its kernel is the intersection of the kernels of the <algebra characters>. For a commutative <C-star algebra>, <spectral radius norm equality for normal elements> and the <Stone-Weierstrass theorem> upgrade this to an isometric <C-star homomorphism> onto $C(\Delta(A))$.
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