Spectral radius norm equality for normal elements
= Spectral radius norm equality for normal elements
{title2=$r(a)=\|a\|$}
For a <Normal element of a C-star algebra>, the <C-star identity> gives $\|a^2\|=\|a\|^2$, and iteration gives $\|a^{2^n}\|=\|a\|^{2^n}$. The <spectral radius formula> then gives $r(a)=\|a\|$. This proves the isometry of the <Gelfand transform> for a commutative <C-star algebra> directly from its defining norm identity.