Do we have $C_0(\mathbb{R}) \cong \{f \in C(S^1): f(1)=0\}$?

Question

This question might be nonsense, but do we have an isomorphism of $C^*$-algebras $$C_0(\mathbb{R}) \cong \{f \in C(S^1): f(1)=0\}$$ where $S^1$ is the unit circle? I can't think of any obvious map from one to the other $C^*$-algebra.

Answer 1

Yes. If $H$ is any homeomorphism of $S^1 \setminus \{1\}$ to $\mathbb{R}$, you get a map of $C^*$-algebras $$\Theta:C_0(\mathbb{R})\rightarrow \{f∈C(S^1):f(1)=0\}$$ defined by $\Theta (g)=g \circ H$.