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Let $K$ be a pseudocompact Tychonoff space that is, a Hausdorff $T_{3.5}$ space for which every continuous function $\varphi \colon K\to \mathbb{C}$ is bounded. Let $\beta$ be the Stone-Cech extension functor. Then

$$\beta\colon C(K)\to C(\beta K)$$

is an algebra isomorphism.

OK, suppose $K$ is moreover locally compact, so one may consider the C*-algebra $C_0(K)$ of continuous functions which vanish at infinity. My question is how large can be in this case the quotient


Can it be infinite dimensional? If so, is there any characterisation of pseudocompactness in terms of relations between $C_0(\beta K)$ and $C(K)$?

PS. By the way, is $\beta$ in this case a *-homomorphism?

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The space $K$ includes into its Stone-Cech compactification, so your map between algebras should go the other direction (a function on $\beta K$ restricts to a function on $K$). – MTS Jul 30 '12 at 21:08
Sure. Corrected. – RadekM Jul 30 '12 at 21:26
@MTS: The other direction $C(\beta K) \to C(K),\; f \mapsto f \circ i: K \to \beta K \to \mathbb{C}$ works as well. – Ralph Jul 30 '12 at 21:46
The quotient algebra $A$ may be infinite dimensional. Consider the space $\omega_{1}\times[0,1]$. The space $\omega_{1}\times[0,1]$ is pseudocompact since every function on this space is eventually constant. In this case, we have $f\in C_{0}(X)$ iff $supp(f)\subseteq\alpha\times[0,1]$ for some ordinal $\alpha$ since every continuous function is eventually constant. Therefore, we have $f+C_{0}(X)=g+C_{0}(X)$ iff $f-g$ is eventually zero. Therefore, the quotient $f+C_{0}(X)$ only considers the tail of the function $f$. Therefore, the algebra $A$ is isomorphic to $C([0,1])$. – Joseph Van Name Jul 31 '12 at 0:20
@Ralph: That's what I meant. I was just regarding $K$ as a subset of $\beta K$, but you're right: it is more properly regarded as a subobject with an inclusion map $i : K \to \beta K$. Then the morphism of algebras $f \mapsto f \circ i$ morally the map given by restriction of functions on $\beta K$ to functions on $K$. – MTS Jul 31 '12 at 3:10

See Example 3.10.29 in Engelking's book (it is due to Katetov): take $X=\beta\mathbb{R}\setminus(\beta\mathbb{N}\setminus\mathbb{N})$. Then $X$ is pseudocompact and in this case quotient algebra $A$ is $C(\beta\mathbb{N}\setminus\mathbb{N})$. See also Problem 3.12.20 in the same book for a result attributed to Van Douwen: if $K$ is compact Hausdorff then $\beta(\omega_1\times K)=(\omega_1+1)\times K$; this shows that every $C(K)$ can occur as your $A$.

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