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.
Best Answer
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.