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.



States with a unique state extension
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.


