= Nonunital continuous functional calculus
{title2=$C_0(\sigma(x)\setminus\{0\})\cong C^*(x)$}
For a <normal C-star element> $x$ in a nonunital algebra, take its <spectrum of an element> in the <C-star unitization>. Zero belongs to that spectrum. Continuous functions on this compact spectrum that vanish at zero form an ideal canonically identified with the displayed function algebra. Their values under the unital <continuous functional calculus> belong to the original algebra because the scalar quotient evaluates them at zero. Star-polynomial approximation with the constant term removed gives exactly the image $C^*(x)$.
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