An isomorphism between two Banach algebras

Question

Consider the compact set $[-1,1]$ and $C([-1,1])$ the set of all continuous functions $\phi: [-1,1] \rightarrow \mathbb{C}$. I want to show that the quotient of $C([-1,1])$ by $\mathbb{C}$ is the Banach algebra $C_0([-1,1])$ of continuous functions $\phi: [-1,1] \rightarrow \mathbb{C}$ that $\phi(0) = 0$. I don't even know if it is true, i need this to prove another result and my intuition say that it is, but i don't know how to formalize this.

Answer 1

"Quotient of $X$ by $Y$" means that $Y$ is somehow thought of as a subset of $X$; a lot depends on how one embeds $Y$ into $X$.

Here's a more concrete way to phrase this problem: you are looking for a surjective homomorphism $f:C[-1,1]\to C_0[-1,1]$ such that the kernel of $f$ is isomorphic to $\mathbb{C}$. This isn't going to be an algebra homomorphism, because the ideals of $C[-1,1]$ are much larger than $\mathbb{C}$; they are infinite dimensional.

A vector space homomorphism isn't hard to find: send each $\phi$ to $\phi-\phi(0)$. I leave it for you to identify the kernel of this homomorphism with $\mathbb{C}$.