maximal ideals of the algebra of continuous functions on a compact set

Let $X$ be a compact Hausdorff space and $C(X)$ the Banach algebra of continuous functions $X⟶ℂ$ (with the sup norm).

In this entry we are interested in identifying the maximal ideals and the character space of $C(X)$. Since $C(X)$ is a Banach algebra with an identity element, there is a bijective correspondence between the character space of $C(X)$ and the set of maximal ideals of this algebra, given by

$$\varphi ⟷Ker\varphi$$

Hence, by identifying the character space of $C(X)$ we are able to identify its maximal ideals.

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Theorem 1 - Let $\mathrm{\Delta }$ be the character space of $C(X)$. For each $x\in X$ let $e{v}_x\in \mathrm{\Delta }$ be the point-evaluation at $x$, i.e.

$$e{v}_x(f)=f(x),f\in C(X)$$

Then the mapping $x⟼e{v}_x$ is an homeomorphism between $\mathrm{\Delta }$ and $X$.

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Thus, the character space of $C(X)$ is homeomorphic to $X$ via point-evaluations.

Now, the maximal ideals of $C(X)$ correspond to the kernels of the point-evaluation functions. The kernel of $e{v}_x$, the point-evaluation at $x$, is just

$\mathrm{\{}f\in C(X):f(x)=0\}$

i.e., the functions that vanish at $x$.

Thus, each maximal ideal of $C(X)$ is just the set of functions that vanish in a given point.

0.1 Generalization to locally compact Hausdorff spaces

Now, let $X$ be a locally compact Hausdorff space and $C_0(X)$ the space of continuous functions $X⟶ℂ$ that vanish at infinity.

There is a generalization of Theorem 1 above that allows one to identify the character space of $C_0(X)$, but since this algebra is not unital unless $X$ is compact, we cannot identify its maximal ideals by the above method.

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Theorem 2- Let $\mathrm{\Delta }$ be the character space of $C_0(X)$. For each $x\in X$ let $e{v}_x\in \mathrm{\Delta }$ be the point-evaluation at $x$, i.e.

$$e{v}_x(f)=f(x),f\in C_0(X)$$

Then the mapping $x⟼e{v}_x$ is an homeomorphism between $\mathrm{\Delta }$ and $X$.

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Thus, the character space of $C_0(X)$ is also homeomorphic to $X$ via point-evaluations.