I am currently attending a course where we are now covering the Stone-Cech compactification. Today we proved in some detail that extensions of bounded smooth functions on $\mathbb{R}^n$ to $\beta\mathbb{R}^n$ are point-separating on $\beta\mathbb{R}^n$. I am finding it a bit difficult to get any motivation as to why anyone would care about this result at all (I guess the corresponding statement for continuous functions would follow trivially from the Urysohn lemma). Does point separation lead to other important consequences? I could really use some insight on this. The professor mentioned in passing about something called the Gelfand-Naimark correspondence. Is that related to this in any way? Thanks!
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6$\begingroup$ Why don't you ask the professor? No doubt they had a definite reason for proving this result and would be happy to explain it. $\endgroup$– Nik WeaverCommented Aug 18, 2015 at 18:42
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$\begingroup$ The existence of a "small'' subspace $U$ of continuous functions separating the points of your space $X$ imposes restrictions on $X$ For example, if $\dim U=n$, then you can continuously embed $X$ in $\mathbb{R}^n$. $\endgroup$– Liviu NicolaescuCommented Aug 19, 2015 at 9:16
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There is, in general, a on-to-one correspondence between closed subalgebras of $C^*(X)$ (the algebra of bounded continuous real-valued functions) and the compactifications of $X$. Closed in the sense of the $\sup$-norm. A compactification $\gamma X$ determines, and is determined by, the closed algebra of members of $C^*(X)$ that extend to $\gamma X$.
If some algebra separates the points of $\beta X$ then by the Stone-Weierstrass theorem it is dense in $C^*(X)$ and in essence determines $\beta X$.
So, the result you quoted implies that $\beta\mathbb{R}^n$ can already be described using the smooth functions only.
