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What is the Stone–Čech compactification $\beta X$ of a compact space $X$, is is $X$ itself? Or does it depend on the definition of compactification, whether the embedding $i:X\to \beta X$ is assumed to be dense or not? See for example here http://ncatlab.org/nlab/show/compactification. This question is similar with this question one-point compactification of a compact space . The background of my question is that in literature is always stated that the Multiplier algebra of $C_0(X)$, $M(C_0(X))$, where $X$ is a locally compact Hausdorff space $X$, is isomorphic to $C_b(X)$ and $C_b(X)$ can be identified with $C(\beta X)$,the continuous functions on the Stone–Čech compactification of $X$. But if $X$ is already compact, then it should be $C_b(X)=C(X)$ but if $\beta X\ncong X$ for compact $X$, you can't identify $C_b(X)$ with $C(\beta X)$ ... But I don't see the mistake here, but I think it depens on the embedding i/ the definition of compactification. Is it correct, or what is the problem here? Greetings

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    For compact $X$, you have $\beta X = X$. It's only for the "one-point-compactification" (also called Alexandrov-compactification) that there's confusion. If we add a single point to a compact space as in the one-point construction, then we don't have a compactification in the strict sense. People often call it that nevertheless. But for $\beta X$, no such confusion exists. – Daniel Fischer Jul 09 '15 at 09:35
  • ok, thanks a lot! – topos Jul 09 '15 at 10:13

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