Extremality of $*$-automorphisms of $C^*$-algebras In a paper, I came across the notion of the extremality of a $*$-automorphism of a $C^*$-algebra. What is the definition of this property?
Edit: Basically, I have a unital completely positive map $\varphi: A \rightarrow A$, where $A$ is a $C^*$-algebra, such that $\psi = (\text{id}_A + \varphi)/2$ is a $*$-automorphism. Then the author states that from this property together with the extremality of a $*$-automorphism, $\varphi = \text{id}_A$, but I don't see how.

The author is saying that automorphisms are extreme among the ucp maps.  
Write $\psi=(\phi_1+\phi_2)/2$, with $\phi_1,\phi_2$ ucp. Let $a\in A$ selfadjoint. Then, as $\phi_1(a)$ and $\phi_2(a)$ are selfadjoint, $(\phi_1(a)-\phi_2(a))^2\geq0$. So
\begin{align}
\psi(a)^2&=\frac{(\phi_1(a)+\phi_2(a))^2}4
=\frac{\phi_1(a)^2+\phi_2(a)^2}2-\frac{(\phi_1(a)-\phi_2(a))^2}4\\ \ \\
&\leq\frac{\phi_1(a)^2+\phi_2(a)^2}2
\leq\frac{\phi_1(a^2)+\phi_2(a^2)}2=\psi(a^2).
\end{align}
Because $\psi$ is multiplicative, $\psi(a)^2=\psi(a^2)$. So the inequalities above become equalities. In particular, we obtain 
$$
(\phi_1(a)-\phi_2(a))^2=0.
$$
Then $\phi_1(a)-\phi_2(a)=0$, since it is selfadjoint. As this works for any selfadjoint $a\in A$, it follows that $\phi_1=\phi_2$.
