# What are the maximal ideals of $C_0 (X),$ where $X$ is a locally compact Hausdorff space?

Crossposted from MSE

How do the maximal ideals of $$C_0(X)$$ look like where $$X$$ is a locally compact Hausdorff space?

I know that if $$X$$ is a compact Hausdorff space then the maximal ideals of $$C(X)$$ are of the form $$I_{c} : = \left \{f \in C(X)\ |\ f(c) = 0 \right \},$$ for some $$c \in X.$$ I can also able to show that if $$X$$ is a locally compact Hausdorff space then $$I_{c}$$ is a maximal ideal of $$C_0 (X)$$ but can't able to prove the converse of that. Could anyone please help me in this regard?

Thanks a bunch.

EDIT $$:$$ Let $$X^+$$ denote the one point compactification of $$X.$$ Let $$\varphi : C_0(X) \longrightarrow \mathbb C$$ be a multiplicative linear functional on $$C_0(X).$$ Then it can be extended to a multiplicative linear functional $$\overline {\varphi} : C(X^+) \longrightarrow \mathbb C$$ defined by $$\overline {\varphi} (f) = \varphi \left (f\ \big |_{X} \right ).$$ Also given any $$f \in C_0(X)$$ there is an unique way to extend it to a function $$\overline {f} \in C(X^+)$$ as follows $$:$$ $$\overline {f} (x) = \begin{cases} f(x) & x \in X \\ 0 & x = \infty \end{cases}$$ Now take any $$f \in C_0(X).$$ Then since $$X^+$$ is compact we have $$\varphi (f) = \overline {\varphi} (\overline {f}) = \overline {f} (c),$$ for some $$c \in X^+.$$ If $$c = \infty$$ we have $$\varphi \equiv 0$$ i.e. $$\varphi$$ is a zero linear functional on $$C_0(X).$$ Otherwise there always exixts some $$c \in X$$ such that $$\varphi (f) = f(c),$$ for all $$f \in C_0(X).$$ So for any non-zero multiplicative linear functional $$\varphi$$ on $$C_0(X)$$ we have $$\text {Ker}\ (\varphi) = \{f \in C_0(X)\ |\ f(c) = 0 \} = I_{c},$$ for some $$c \in X.$$ Since we know that maximal ideals in any Banach algebra $$A$$ are precisely the kernels of non-zero multiplicative linear functionals on $$A$$ and $$C_0(X)$$ is a Banach algebra (in fact a $$C^{\ast}$$-algebra) we are through.

Possible Mistake $$:$$ I think where my above argument goes wrong is the stage where I assume $$f\ \big |_{X} \in C_0 (X)$$ whenever $$f \in C(X^{+}).$$ Instead when $$f \in C_0 (X^+)$$ this happens. Now we need to search for maximal ideals of $$\{f \in C(X^+)\ |\ f(\infty) = 0 \}.$$ So the above is not a valid argument. What my above argument shows is the following $$:$$

Multiplicative linear functionals on $$C(X)$$ are precisely evaluations even if $$X$$ is locally compact.

So the original question ultimately boils down to the following question $$:$$

Can we always extend every multiplicative linear functional on $$C_0(X)$$ to a multiplicative linear functional on $$C(X)$$ if $$X$$ is a locally compact Hausdorff space?

• What is $C_0(X)$? Functions with compact support or functions going to zero at infinity? Commented Oct 17, 2021 at 19:36
• If $\varphi$ is a multiplicative linear functional on $C_0(X)$, then isn't $\varphi^+ : c + f \mapsto c + \varphi(f)$ a multiplicative linear functional on $C(X^+)$, where we write $f$ also for the extension of $f$ to $X^+$ that vanishes at $\infty$, and $c$ also for the constant function on $X^+$ with value $c$? Then $\varphi^+$ is the evaluation at some point $x \in X^+$ ($x = \infty$ if and only if $\varphi = 0$), and $\ker(\varphi) = \ker(\varphi^+) \cap C_0(X)$. Commented Oct 17, 2021 at 20:26
• Although the question has received answers, or outlines of answers, here, it belongs more properly on MSE. It has only been on MSE for less than 24 hours; in future, perhaps try waiting a bit longer before crossposting here. Commented Oct 18, 2021 at 2:18
• Oh right. Since kernels of non-zero multiplicative linear functionals are necessarily maximal ideals of the underlying Banach algebra we are through. But how do I guarantee that such an evaluation is always non-zero? In other words given $c \in X$ can we always find some $f \in C_0(X)$ such that $f(c) \neq 0$ @PietroMajer? For specific $X ( = (0,1)$ (say)) it is easy to see. But for arbitrary $X$ (LCH) is it always true?
– RKC
Commented Oct 19, 2021 at 8:24
• Because X was assumed to be a locally compact Hausdorff space, hence completely regular (so for instance, for any c and any nbd U of c, there are functions with f(c)=1 and vanishing outside U). en.wikipedia.org/wiki/Tychonoff_space Commented Oct 19, 2021 at 8:35