(Reference request) Can real-valued "characters" on the algebras of continuous/smooth/analytic functions always be expressed as point evaluations? Let $M$ be a manifold with possible boundary (but not necessarily compact). We may assume $M$ is topological, smooth, analytic, etc. as necessary, throughout.
Let $A$ be the (assoc,comm,unital) real algebra of continuous/smooth/analytic functions on $M$.
Are all (real-linear?) ring homomorphisms $A \rightarrow \mathbb{R}$ given by $\mathrm{eval}_p$ for some $p \in M$?
If not, does this become true if we require only continuous homomorphisms, with respect to some choice of topology on $A$ (making it a topological algebra)?
If I'm not mistaken many of these questions are answered in analysis (of function spaces), but I'm not familiar with the literature. Could anyone suggest a reference which discusses the above questions in detail?
(Note: I have added a "Banach algebras" tag although I'm not sure when the above function spaces have a Banach algebra description.)
 A: This is equivalent to asking whether every maximal ideal of $A$ with residue field $\mathbb{R}$ is of the form $\mathfrak{m}_p := \ker \operatorname{eval}_p = \{f\in A:\, f(p) = 0\}$ for some $p\in M$. For $M$ compact, this is true (at least in reasonable categories of functions); in fact, every maximal ideal is of that form, and they all have residue field $\mathbb{R}$. For if $\mathfrak{m} < A$ is maximal with $\mathfrak{m} \not = \mathfrak{m}_p$ for any $p$, then there exists for each $p\in M$ some $f_p\in \mathfrak{m}$ with $f_p(p)\not = 0$. Since $f_p$ is continuous, there exists some open neighborhood $U_p$ of $p$ with $f_p\vert U_p$ nonzero. The $U_p$ admit a finite subcover $U_{p_1}, \dots, U_{p_n}$ by the compactness of $M$. But then $F = f_{p_1}^2 + \cdots + f_{p_n}^2\in \mathfrak{m}$ is nonzero everywhere, so $1 = (1/F)F\in \mathfrak{m}$ as well. Oops.
Under some reasonable assumptions on the topology of $M$, the result also holds in the noncompact case. See here and the book "$C^\infty$-differentiable Spaces" referenced there.
