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.