= Unital versus nonunital generation by a normal element
{title2=$C^*(1,x)\text{ versus }C^*(x)$}
The unital <continuous functional calculus> has image $C^*(1,x)$, the <norm> closure of star polynomials including constants. The smallest closed star subalgebra $C^*(x)$ need not contain the ambient identity: for $x=0$ it is zero. In a nonunital algebra, apply the unital calculus in the <C-star unitization> and retain functions vanishing at zero to recover $C^*(x)$ inside the original algebra.
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