= Solution
The <unitization of an algebra>[unitization] of $C_0(\mathbb R)$ is $C(\mathbb R^+)$ on the <one-point compactification> of $\mathbb R$. The supplied homeomorphism identifies $\mathbb R^+$ with the <circle> $\mathbb T$. Every character $\varphi$ of $C_0(\mathbb R)$ extends to the unital character
$$
\widetilde\varphi(f+\lambda1)=\varphi(f)+\lambda.
$$
By part d, $\widetilde\varphi$ is evaluation at a point of $\mathbb R^+$. Evaluation at the point at infinity vanishes on $C_0(\mathbb R)$ and cannot restrict to the nonzero character $\varphi$. The point is therefore some $x\in\mathbb R$, and $\varphi=\delta_x$. Conversely each $\delta_x$ is plainly a character, so
$$
\Phi_{C_0(\mathbb R)}=\{\delta_x:x\in\mathbb R\}.
$$
Solved by gpt-5.6-sol high.
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