# Determining non-zero multiplicative linear functionals on $C_0(X).$

banach-algebrasc-star-algebrasoperator-algebras

Let $$X$$ be a locally compact, non-compact and Hausdorff space. Then I know that the closed maximal ideals of $$C_0(X)$$ are the kernels of some evaluations. Also I am familiar with the fact that the closed maximal ideals in a Banach algebra are precisely kernels of non-zero multiplicative linear functionals on it. That means kernels of non-zero multiplicative linear functionals on $$C_0(X)$$ are kernels of some evaluations. Does it imply that all the non-zero multiplicative linear functionals on $$C_0(X)$$ are themselves evaluations? To be more explicit I want to know the following $$:$$

Given a non-zero multiplicative linear functional on $$C_0(X)$$ can I say that $$\varphi$$ is nothing but an evaluation if it is known that $$\text {ker}\ \varphi = I_{c} : = \{f \in C_0(X)\ |\ f(c) = 0\},$$ for some $$c \in X.$$

The problem I am facing due to the fact that $$C_0(X)$$ is non-unital since $$X$$ is non-compact. Otherwise the argument would be simple. For any $$f \in C_0(X)$$ we consider the function $$g : = f – f(c)\ 1.$$ Then $$g(c) = 0.$$ But then $$g \in \text {ker}\ \varphi$$ and hence $$\varphi (f) = f(c),$$ as required. But I don't know how to tackle the same situation for non-unital case. Could anyone give me some suggestion regarding this?

Thanks a bunch.

This is what you already came up with (and is also how it is shown in Davidson's C*-Algebras By Example): if $$\varphi$$ is a multiplicative functional on $$C_0 (X)$$, extend it to a functional on $$C (\widetilde X)$$, where $$\widetilde X$$ is the one-point compactification of $$X$$. We can do this by writing elements of $$C (\widetilde X)$$ as $$f + \lambda I$$ for $$f \in C_0(X)$$ ($$\lambda$$ will be their value at the point at infinity) and setting $$\widetilde \varphi (f+ \lambda I) = \varphi(f) + \lambda$$ This is linear and is multiplicative because $$\widetilde \varphi ((f_1 + \lambda_1 I) (f_2 + \lambda_2 I)) = \widetilde \varphi (f_1 f_2 + \lambda_1 f_2 + \lambda_2 f_1 + \lambda_1 \lambda_2 I) =$$ $$= \varphi (f_1 f_2 + \lambda_1 f_2 + \lambda_2 f_1) + \lambda_1 \lambda_2 = \left( \left( \varphi (f_1) + \lambda_1 \right) \left( \varphi (f_2) + \lambda_2 \right) \right)$$ et cetera.