Unitization of the vanishing-at-zero ideal on an interval ($\{f\in C[0,1]:f(0)=0{\}}^{\sim }\cong C[0,1]$)

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 $(f,\lambda )\mapsto f+\lambda 1$ 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.