= Unitization of the vanishing-at-zero ideal on an interval
{title2=$\{f\in C[0,1]:f(0)=0\}^{\sim}\cong C[0,1]$}
Evaluation at zero is a bounded homomorphism on $C[0,1]$. Its kernel $I$ is a closed self-adjoint ideal and hence a <C-star subalgebra>. It has no identity: applying a proposed identity to $f(t)=t$ would force that identity to equal one at every positive point, contradicting continuity and its value zero at zero. The map $(f,\lambda)\mapsto f+\lambda\mathbf1$ from the <unitization of a C-star algebra> is a bijective star homomorphism, with inverse $h\mapsto(h-h(0),h(0))$. Pulling back the <supremum norm> verifies the <C-star identity>, so this is the prescribed <unitization> <norm>. Thus adjoining the missing constant direction recovers all <continuous functions> on the interval.
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