# Function spaces over pseudocompact spaces

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

$$A:=C(K)/C_0(K)?$$

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?

• 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 $A$.