# 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 $X\longrightarrow\mathbb{C}$ (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

 $\phi\longleftrightarrow Ker\;\phi$

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

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

 $ev_{x}(f)=f(x)\;,\qquad\qquad\;f\in C(X)$

Then the mapping $x\longmapsto ev_{x}$ is an homeomorphism between $\Delta$ 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 $ev_{x}$, the point-evaluation at $x$, is just

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

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 $X\longrightarrow\mathbb{C}$ 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 $\Delta$ be the character space of $C_{0}(X)$. For each $x\in X$ let $ev_{x}\in\Delta$ be the point-evaluation at $x$, i.e.

 $ev_{x}(f)=f(x)\;,\qquad\qquad\;f\in C_{0}(X)$

Then the mapping $x\longmapsto ev_{x}$ is an homeomorphism between $\Delta$ and $X$.

$\;$

Thus, the character space of $C_{0}(X)$ is also homeomorphic to $X$ via point-evaluations.

Title maximal ideals of the algebra of continuous functions on a compact set MaximalIdealsOfTheAlgebraOfContinuousFunctionsOnACompactSet 2013-03-22 17:44:57 2013-03-22 17:44:57 asteroid (17536) asteroid (17536) 5 asteroid (17536) Theorem msc 46L05 msc 46J20 msc 46J10 msc 16W80 character space of the algebra of continuous functions on a compact set