Following Bredon's Topology and Geometry, we let $\mathcal{F}$ be the set of all continuous maps $f:X \to [0,1]$ on a completely regular space $X$, define $X \xrightarrow{\Phi} [0,1]^{\mathcal{F}}$ by setting $\Phi(x)(f)=f(x)$ for each $x \in X$ and $f \in \mathcal{F}$, and declare the closure $\beta(X)$ of $\Phi(X)$ in $[0,1]^{\mathcal{F}}$ to be the Stone-Čech compactification of $X$. Munkres writes in Topology that there are "a number of applications [of the Stone-Čech compactification] in modern analysis", which is purportedly "outside the scope of [the] book." I looked at a few sources, including The Stone-Čech compactification by Russell Walker, and failed to find any application of the theorem that is overtly analytic. Perhaps I do not have sufficient background in functional analysis—this is where, I assume, the prototypical analytic applications would be in–to recognize the functional-analytic applications I have encountered, but the point remains that I have yet to see such an example. So:
What are prototypical applications of the Stone-Čech compactification in mathematical analysis, and where can I read about them?
Best Answer
I'm not sure this is what Munkres is talking about, but here's something he could be talking about. Let $X$ be a completely regular space and let $C_b(X)$ be the ring of bounded continuous functions $X \to \mathbb{R}$. This is a commutative Banach algebra when equipped with the sup norm, and in fact it is a $C^{\ast}$-algebra when equipped with the trivial involution.
Then the Gelfand spectrum of $C_b(X)$ is canonically isomorphic to $\beta X$; equivalently, $C_b(X)$ is canonically isomorphic to $C(\beta X)$. One application here is that by the Riesz representation theorem, positive linear functionals on $C_b(X)$ can be identified with Borel regular measures on $\beta X$. A special case of this construction is described in the Wikipedia article.