Question

This question comes up after I read the chapter 7 : Banach algebras and spectrum theory from Conway's book. As we have known that if $X$ is compact space, then all the maximal ideals of $C(X)=\{ f : X\to \mathbb{C}\}$ are of the forms $m_x =\{f\in C(X) : f(x)=0\}$. Now assume that $X$ is just locally compact and let $X^{+}$ and $\overline{X}$ be the one-point compactification, and Stone-Čech compactification of $X$ respectively. Let $I_{0}(X)$, $I(X^+), I(\overline{X})$ be the set of all maximal ideals of $C_{0}(X), C(X^+), C(\overline{X})$ respectively. My question here is that: do we have any relation between these sets $I_{0}(X)$, $I(X^+), I(\overline{X})$ ? For example, if we know $I(X^+)$, how can we find $I_0(X)$ (because if we know the answer for this, it seems for me that one proof of the Stone-Čech theorem could be found from here) ?

Answer 1

The set $I(X^+)$ is just the set of all points of $X^+$, and similarly the set $I(\overline{X})$ is the set of points of $\overline{X}$.

On the other hand the set $I_0(X)$ is somewhat mysterious. In general the ideals of a commutative but nonunital Banach algebra are harder to deal with.

When $X$ is not compact, the maximal ideals of $C_0(X)$ are more than just the points of $X$. The points of $X$ are in 1-1 correspondence with the kernels of (nonzero) characters $C_0(X) \to \mathbb{C}$. These are known as modular ideals (see Exercises 6 and 7 in $\S VII.2$ of Conway), or also known as regular ideals, see e.g. Exercise 4.2.1 of the excellent book Analysis Now, by Gert Pedersen.

Too see the sort of thing that goes wrong in the nonunital situation, for instance let $$ M = \{ f \in C_0(X) \mid \mathrm{supp}(f) \text{ is compact} \}. $$ Then certainly $M$ is an ideal. But no evaluation functional vanishes on $M$, and in fact $M$ is dense in $C_0(X)$.

In a unital Banach algebra, proper ideals are always contained in maximal ones, but this shows that the same is not the case in the nonunital situation. I don't know if this $M$ is actually contained in a maximal proper ideal (which would necessarily be non-closed) but if so it would certainly not correspond to a point of $X$ and I imagine any such ideal would be impossible to describe in a constructive way.

Sorry if this was a bit incoherent. It's been a long time since I've thought about these things. I don't think I actually answered your question but hopefully it's given you something to think about.