I'm a little embarassed that I can't answer this myself, so hopefully it will get answered very quickly.

Let $X$ be locally compact, Hausdorff. Consider $\text{C}_\text{b}(X)$ the $C^*$-algebra of bounded, complex-valued, continuous functions on it under the sup norm. The Gelfand transform takes $\text{C}_\text{b}(X)$ isomorphically to $\text{C}_0(Y)$ where $Y$ is the maximal ideal space of $\text{C}_\text{b}(X)$. Since $\text{C}_\text{b}(X)$ is unital, $Y$ must be compact.

It is well known that $Y$ is the Stone–Čech compactification of $X$. If $X = \mathbb{R}$ then $Y$ is the one-point-compactification. In general, the maximal ideal space is in one-to-one correspondence with the characters (nonzero multiplicative linear functionals) on $\text{C}_\text{b}(X)$.

Question: Say $X = \mathbb{R}$ and the maximal ideal space of $\text{C}_0(\mathbb{R})$ is made up of the point evaluations at points on $\mathbb{R}$. These point evaluations are also characters on $\text{C}_\text{b}(X)$, but its maximal ideal space is the one-point-compactification of $\mathbb{R}$. What character on $\text{C}_\text{b}(\mathbb{R})$ represents this single missing point of the maximal ideal space?

The big issue I'm having is that for a non-compact space like $\mathbb{R}$, we have an obvious problem that you could take a sequence of point evaluations $\chi_n(f) = f(n)$ where $n \to \infty$, which clearly goes to the zero functional if your $f \in \text{C}_0(\mathbb{R})$. This doesn't happen in $\text{C}_\text{b}(\mathbb{R})$ because the maximal ideal space doesn't need to add the zero functional in order to be compact, but that would imply that $\chi_n$ is converging to some point evaluation "at infinity" (which fits with the idea that $\mathbb{R}$ gets compactified by adding a point "at infinity"). But $\chi_n(f) = f(n)$ as $n\to\infty$ still doesn't make sense for functions $f \in \text{C}_\text{b}(\mathbb{R})$ (take $\sin(x)$ for example).

  • 1
    $\begingroup$ Maybe I'm misreading you, but the Stone-Čech compactification of $\mathbb{R}$ can't possibly be the one-point compactification, since $C_b(\mathbb{R})$ isn't isomorphic to $C_0(\mathbb{R})^+ \cong C_0((0,1))^+ \cong C(S^1)$. $\endgroup$ – Branimir Ćaćić Feb 16 '14 at 0:46
  • $\begingroup$ Are you sure that the Stone–Čech compactification of $\mathbb{R}$ is the one-point compactification? This seems to contradict the universal property stated here: en.wikipedia.org/wiki/… For example, not all continuous functions from $\mathbb{R}$ to $[0,1]$ factor through a map from $S^1$. $\endgroup$ – Paolo Ghiggini Feb 16 '14 at 0:47
  • $\begingroup$ The Stone-Čech compactification is defined for any topological space, with no hypotheses whatsoever. The map $X \to \beta X$ is an embedding iff $X$ is Tychonoff, which is still a weaker condition than $X$ being locally compact Hausdorff (this is the condition you need for the existence of the one-point compactification). $\endgroup$ – Qiaochu Yuan Feb 16 '14 at 4:58
  • 2
    $\begingroup$ The confusion in your question has been cleared up by Qiaochu Yuan but the following remark might interest you. The compactifications of $X$ correspond to the $C^\ast$ subalgebras of $C^b(X)$ ($X$ need not be locally compact). Interesting special cases: the bounded uniformly continuous functions (for uniform spaces), the algebra obtained by adding a unit to $C_0(X)$ ($X$ locally compact), the almost periodic functions ($X$ a locally compact group) and many more. The corresponding spectra are, in these cases, the Samuel, the one-point and the Bohr compactification. $\endgroup$ – alpha Feb 16 '14 at 10:43

The Stone-Čech compactification of $\mathbb{R}$ is not its one-point compactification. The former is the largest compactification of a space, while the latter, if it exists, is the smallest compactification, and in general there will be many compactifications in between. $\mathbb{R}$ has, for example, a two-point compactification, namely $[0, 1]$.

The C*-algebraic construction of the one-point compactification is not $C_b(\mathbb{R})$ but the subalgebra

$$\{ f \in C_b(\mathbb{R}) | \lim_{x \to \infty} f(x) = \lim_{x \to -\infty} f(x) \}$$

and taking the limit is the character corresponding to the extra point. (In general, if $X$ is locally compact Hausdorff, then we can construct its one-point compactification by adjoining a unit to $C_0(X)$.)

Requiring only that the limits exist but not that they agree gives the two-point compactification. More generally, points in the Stone-Čech compactification can be described in terms of ultralimits.


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.