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I am having some trouble constructing the Stone-Čech compactification of a locally compact Hausdorff space $X$ using theory of $C^*$-algebras. I did some search but could not find a good answer on this.

Let's focus on the case $X=\mathbb{R}$. The space of bounded complex-valued functions $C_b(\mathbb{R})$ is a commutative unital $C^*$-algebra hence $C_b(\mathbb{R})\cong C(\mathcal{M})$, where $\mathcal{M}$ is the maximal ideal space, which is compact and Hausdorff.

It should be the case that $\mathcal{M}\cong\beta\mathbb{R}$, and it is not difficult to show that by identify $t\in\mathbb{R}$ with the evaluation at $t$, we have a homeomorphism between $\mathbb{R}$ and a subspace of $\mathcal{M}$.

But we still need to show this subspace is dense in $\mathcal{M}$. This is where I am having troubles (and I guess this is the whole point of the proof).

Can someone give a hint? Thanks!

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    Curious question: why would you like to construct the SC-compactification using C-star theory?2012-12-17
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    @MattN. Well, I was looking at Stone-Cech on Wiki, which mentions this approach in one sentence. So I tried to give a proof.2012-12-17

2 Answers 2

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The sketch in the other answer takes care of much, except that it doesn't address the question why it is that $i(X)$ is dense in $\mathcal{M}$.

So: let $i \colon X \to \mathcal{M}$ be the map sending $x$ to (the maximal ideal corresponding to) evaluation at $X$. If $i(X)$ were not dense then there would be a function $f \colon \mathcal{M} \to [0,1]$ such that $f|_{i(X)} = 0$ (apply Urysohn's lemma to a point outside of the closure of $i(X)$). But the existence of such a function is impossible since such a function would have to be zero under the identification $C(\mathcal{M}) \cong C_b(X)$.

A detailed proof of the Stone-Čech property of the maximal ideal space of $C_b(X)$ appears in many books treating spectral theory of $C^\ast$-algebras, e.g. Pedersen, Analysis now, Proposition 4.3.18.

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    Yes. I found the proof from Pederson earlier today. Thanks!2012-12-18
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You really should be thinking about the Stone-Cech compactification in terms of its universal property; the inclusion $X \to \beta X$ is already uniquely determined (up to unique isomorphism) by the fact that it is the universal map from $X$ to a compact Hausdorff space, so to verify that $C_b(X) \cong C(\beta X)$ it suffices to verify that the compact Hausdorff space $Y$ such that $C_b(X) \cong C(Y)$ (which exists by Gelfand-Naimark) has the universal property of the Stone-Cech compactification.

(There is also no need to assume that $X$ is locally compact Hausdorff. Everything I'm about to say makes sense for arbitrary topological spaces, although the map $X \to \beta X$ is only an embedding for $X$ completely regular.)

To verify the universal property, let $f : X \to Z$ be a continuous map from $X$ to a compact Hausdorff space $Z$. Then $f$ determines a map $C(Z) \to C_b(X)$ of C*-algebras (a complex-valued function on $Z$ will be bounded, and so its pullback to $X$ will also be bounded). Since $C_b(X) \cong C(Y)$, it follows that $f$ determines a map $C(Z) \to C(Y)$, and by the equivalence of categories between commutative unital C*-algebras and compact Hausdorff spaces (this is the technical heart of the proof) this uniquely determines a continuous map $Y \to Z$ through which $f$ factors. The conclusion follows.

Edit: The fact that (the image of) $X$ is dense in $\beta X$ follows directly from the universal property, since the closure of $X$ in $\beta X$ satisfies the universal property of the Stone-Cech compactification, hence its inclusion into $\beta X$ must be an isomorphism.

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    What you said is true. But the universal property assumes the existence of $\beta X$ in the first place, while the construction using $C_b(X)\cong C(\mathcal{M})$ *constructs* $\beta X$ as the maximal ideal space hence proves the existence. If you already assumes the existence of Stone-Cech (from other constructions, maybe) then universal property gives a nice proof, but the other answer has the advantage that it assumes less. But thanks anyway.2012-12-18
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    @Hui: I don't understand what you're saying. You seem to be criticizing this proof but I do not see where the criticism is. $\beta X$ is constructed as the Gelfand spectrum of $C_b(X)$ and the rest is verifying the universal property, which is how we know that we have actually constructed $\beta X$ and not something else.2012-12-18
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    There is nothing wrong with your proof, and sorry if I made you feel I am criticizing. However, if someone knows nothing about S-C compactification, it seems more natural to first construct a compactification, then verify the universal the universal property, not the other way around. Tell me if I am mistaken, but I always feel universal properties are very neat and useful for understanding, they always come after a concrete construction. I am not sure about S-C.2012-12-19
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    Did they first construct the compactification then discover the universal property? Or did they first ask 'we want something with this universal property, can we construct it'?2012-12-19
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    @Hui: universal properties are something you can demand and prove things about completely independent of any construction of objects satisfying them. That is what makes universal properties a powerful tool: because objects satisfying universal properties unique up to unique isomorphism, anything you want to prove about a universal construction should follow _from its universal property alone_ (in particular the claim that $X$ is dense in $\beta X$). Thinking in terms of universal properties separates all of these issues from the issue of existence (that is, construction).2012-12-19
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    @Hui: I don't know what the historical order of events is, but it's natural from a modern perspective to ask for an object satisfying the universal property and then to construct it, which as I understand it can be done using a suitable version of the adjoint functor theorem.2012-12-19
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    @Hui: I am also not even sure what it would mean to do things in the other order. If you didn't know what you were looking for, why would you be motivated to look for it?2012-12-19
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    @Hui: finally, I don't think your criticism even applies to my proof. This proof constructs the Stone-Cech compactification as the Gelfand spectrum of $C_b(X)$, then verifies that it has the universal property.2012-12-19
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    Again, I am not criticizing your proof. You and I have very different tastes about universal properties and that is just fine. As for 'if you did not know what you were looking for, why would you be motivated to look for it?', it seems to me quite sometime we did not know what we were looking for until I found them.2012-12-19
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    @Hui: yes, but that doesn't mean it's either a good way to think about the things we've found or that it's a good way to find things in the future. Penicillin was also discovered by accident but we don't teach scientists how to find things by accident.2012-12-19
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    :) That's a good one!2012-12-19