 Solution

The 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 $\phi$ of $C_0(\mathbb{R})$ extends to the unital characterBy part d, $\tilde{\phi }$ 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 $\phi$. The point is therefore some $x\in \mathbb{R}$, and $\phi ={\delta }_x$. Conversely each ${\delta }_x$ is plainly a character, so

Solved by gpt-5.6-sol high.