Suppose we have a $C^*$ algebra $\mathfrak{U}$ that is non-separable. Consider a state $ω$ of $\mathfrak{U}$ and the GNS representation $(H_ω,π_ω,Ω)$. Is it possible for $H_ω$ to be separable, and if yes under what assumptions?

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    $\begingroup$ Yes, take $\ell^\infty(\mathbb{N})$. It is not separable, and then the state $\text{ev}_n$ yields a 1-dimensional GNS. The answer to your question is yes even in the case for some state of full support as well. $\endgroup$ Commented Dec 4, 2023 at 17:57
  • $\begingroup$ Okay, I thought this would be the case. Thanks for the example! Is there a relevant theorem, stating assumptions under which the Hilbert space will be separable? Or is this not an interesting problem? $\endgroup$
    – Arbiter
    Commented Dec 4, 2023 at 18:02
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    $\begingroup$ I’m not sure there are interesting general criteria other than just saying the associated Hilbert space is separable. Maybe one criteria if you study von Neumann algebras is that if your $C^*$ algebra is actually a separable vNa (which is nearly never separable as a $C^*$ algebra), and if your state is normal, then the GNS Hilbert space would be separable. $\endgroup$
    – David Gao
    Commented Dec 4, 2023 at 18:19


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