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Let $X$ be a locally compact second countable space. Consider the $C^*$- algebra $C_0(X)$. Is it true that there is a one to one correspondence between states on $C_0(X)$ and Borel probability measures on $X$? Or is it true only $X$ is compact? In that case $C_0(X)=C(X)$. I have $Y$ is a compact subspace of $X$ and I have a Borel probability measure on $Y$. Can I extend it to $X$?

budi
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Yes, it is true. There is a one to one correspondence between states on $C_0(X)$ and Radon probability measures. This is the content of the Riesz representation theorem. However, we also have the following theorem:

If $X$ is a locally compact Hausdorff space that is second countable, then every finite Borel measure on $X$ is Radon. See Folland's "Real analysis" theorem 7.8.

J. De Ro
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  • What happens if $X$ is compact? Still it corresponds to Radon Probability measures? – budi Aug 24 '20 at 11:08
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    Yes, if $X$ is compact (Hausdorff), then $X$ is automatically locally compact (Hausdorff) as well so the results still holds. – J. De Ro Aug 24 '20 at 11:11
  • How can I do the extension of a Borel probabilty measure defined on a compact subspace $Y$ of $X$ to $X$? Is it possible? – budi Aug 24 '20 at 11:13
  • I'm not quite sure about that. Sorry. – J. De Ro Aug 24 '20 at 11:52
  • @AAh I was mistaken when I wrote my answer. I missed the second countability hypothesis. This ensures that Borel measures are automatically Radon when they are finite on compact sets. – J. De Ro Aug 24 '20 at 12:22