Function spaces over pseudocompact spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:05:12Z http://mathoverflow.net/feeds/question/103548 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103548/function-spaces-over-pseudocompact-spaces Function spaces over pseudocompact spaces RadekM 2012-07-30T20:38:19Z 2012-09-01T14:22:00Z <p>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</p> <p>$$\beta\colon C(K)\to C(\beta K)$$</p> <p>is an algebra isomorphism.</p> <p>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</p> <p>$$A:=C(K)/C_0(K)?$$</p> <p>Can it be infinite dimensional? If so, is there any characterisation of pseudocompactness in terms of relations between $C_0(\beta K)$ and $C(K)$?</p> <p>PS. By the way, is $\beta$ in this case a *-homomorphism?</p> http://mathoverflow.net/questions/103548/function-spaces-over-pseudocompact-spaces/103944#103944 Answer by KP Hart for Function spaces over pseudocompact spaces KP Hart 2012-08-04T13:14:41Z 2012-08-04T13:14:41Z <p>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 $A$. </p>