I guess that the answer to the following question is both well known and easy. But I was unable to solve the exercise.

Consider a unital $C^*$-$\,$algebra $\mathcal A$ and and a proper unital sub-$C^*$-$\,$algebra $\mathcal B\subset\mathcal A$. Let also $\varphi$ be a state on $\mathcal B$. Assume that $\varphi$ has a unique state extension to $\mathcal A$. Is the state $\varphi$ necessarily pure?

Apologies if this has already been asked (possibly several times); and thanks in advance for any answer and/or reference.


No; consider $\mathbb C \oplus \mathbb C \oplus \mathbb C \subset \mathbb C \oplus M_2(\mathbb C)$ (in the obvious way) with the state $\varphi(x_1,x_2,x_3)= \frac12(x_1 + x_2)$. Then, the extension of $\varphi$ is unique and $\varphi$ is not pure.

  • $\begingroup$ That's fine, many thanks! Just for curiosity's sake, would you know any general condition on the pair $(\mathcal B,\mathcal A)$ ensuring that the answer is "Yes"? $\endgroup$ – Etienne Nov 8 '14 at 19:06
  • $\begingroup$ Maybe $\mathcal A$ simple, but I don't know. $\endgroup$ – Andreas Thom Nov 8 '14 at 19:08

Yes; the set of extensions of a closed face (in this case a singleton) is itself a closed face.

  • $\begingroup$ I'm afraid I don't understand your answer. $\endgroup$ – Etienne Nov 8 '14 at 14:22
  • $\begingroup$ More precisely, I think that you answered another question, namely "if a pure state has a unique state extension, is this extension necessarily pure?"; which is, of course, true. $\endgroup$ – Etienne Nov 8 '14 at 16:49
  • $\begingroup$ Yes, I misread the question. $\endgroup$ – David Handelman Nov 8 '14 at 23:23

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