Timeline for When is it $C(X)$?
Current License: CC BY-SA 3.0
18 events
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| Feb 28, 2014 at 1:18 | comment | added | Bill Johnson | Continuing: Maharam's theorem gives a characterization of $L_1(\mu)$ spaces, but which ones of these are isometric to dual spaces? The answer could be easy, but is unknown to me. | |
| Feb 28, 2014 at 1:14 | comment | added | Bill Johnson | I like this formulation of your question better. Suppose $Y$ is $\mathcal{L}_\infty$ $1+\epsilon$ for every $\epsilon >0$. Is there a compact Hausdorff space $K$ s.t. $C(K)^*$ is isometrically isomorphic to $Y^*$? Lacey showed that the answer is yes if $Y$ is separable. I have no idea what the answer is when the density character of $Y$ is uncountable. | |
| Feb 27, 2014 at 15:37 | answer | added | M.González | timeline score: 6 | |
| Feb 27, 2014 at 15:26 | comment | added | M.González | @Yemon Choi: Yes. He only proves isomorphic. This is the reason I did not put it as an answer. I will put a more detailed version as an answer. | |
| Feb 27, 2014 at 14:53 | comment | added | Yemon Choi | @M.González Thanks - I wasn't aware of Haydon's paper. Am I right in thinking that he only proves "isomorphic" and not "isometrically isomorphic"? Regardless, you should post your comment (perhaps with some added references) as an answer | |
| Feb 27, 2014 at 7:39 | comment | added | M.González | If $K$ is a compact space and $C(K)$ is isomorphic to a second dual, then it is injective and, by a result of Haydon, it is isomorphic to $\ell_\infty(\Gamma)$ for some set $\Gamma$. Hence $C(K)$ is isomorphic to $c_0(\Gamma)^{**}$. | |
| Feb 27, 2014 at 4:33 | comment | added | Bob | Also, since $C_0(X)$ has an approximate identity, $C_0(X)^{**}$ must have a unit. So all these together implies that $C_0(X)^{**}$ is a unital commutative $C^*-$algebra. | |
| Feb 27, 2014 at 4:32 | comment | added | Bob | A reference can be "Seever, G. L., Algebras of continuous functions on hyper-stonian spaces, Arch. Math. (Basel) 24 (1973), 648–660". Very briefly, if you have a Banach algebra then you can make its second dual into a Banach algebra using Arens product. Now it can be shown that the second dual of a $C^*-$algebra is again a $C^*-$algebra with the Arens product. You can also show that in the case of $C(X)$, the Arens product on $C(X)^{**}$ is commutative (please, see the end of page 8 of the above paper). | |
| Feb 27, 2014 at 4:27 | comment | added | Paul Siegel | OK, so the examples of spaces $X$ that we're talking about are god-awful then... | |
| Feb 27, 2014 at 4:26 | history | edited | Yemon Choi |
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| Feb 27, 2014 at 4:26 | answer | added | Yemon Choi | timeline score: 6 | |
| Feb 27, 2014 at 4:21 | comment | added | Yemon Choi | @PaulSiegel just take an abelian von Neumann algebra. Diffuse would get you a dual space, atomic would get you a bidual space. | |
| Feb 27, 2014 at 4:18 | comment | added | Paul Siegel | Let's make things more concrete. Consider the case $X = [0,1]$, and suppose $C(X)$ were the dual of a Banach space. By Krein-Milman the unit ball of $C(X)$ would be the closed convex hull of its extreme points, but I claim the constant functions $1$ and $-1$ are the only extreme points. Indeed, suppose $f$ is a continuous function on $[0,1]$ with max value $1$, but suppose $f(x) < 1$ for some $x$. Choose a bump function $g$ of very small height which is supported in a very small neighborhood of $x$, and observe that $f$ is the convex combination of $f + g$ and $f-g$. | |
| Feb 27, 2014 at 4:11 | comment | added | Paul Siegel | What is your proposed C*-algebra structure on $C(X)^{**}$? | |
| Feb 27, 2014 at 3:51 | comment | added | Bob | I don't see why that's true. Here is an example. If $X$ is a locally compact space, then $C_0(X)^{**}$ is a unital commutative $C^*-$algebra and therefore it is in the form of $C(\tilde{X})$ for a compact space $\tilde{X}$. This ($C(\tilde{X})$) gives an example of a $C(X)$ that is a dual of a Banach space. | |
| Feb 27, 2014 at 3:46 | history | edited | Bob | CC BY-SA 3.0 |
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| Feb 27, 2014 at 3:35 | comment | added | Paul Siegel | I'm pretty sure that $C(X)$ can't be the dual of any Banach space by the Krein-Milman theorem... | |
| Feb 27, 2014 at 3:22 | history | asked | Bob | CC BY-SA 3.0 |