extreme vector in the positive cone of $C*$-algebra

Question

The definition of extreme vector is defined in the screenshot.Suppose $A$ is a unital $C^*$-algebra,$A^{+}$ is the set of all positive elements in $A$,does there exist an extreme vector in $A^{+}$?

Answer 1

An $x$ satisfying your definition cannot be invertible if it's not a scalar (you have $x^{-1}\leq\|x^{-1}\|$, so $\|x^{-1}\|^{-1}\leq x$), so $0\in\sigma(x)$.

By looking at $C^*(x)\simeq C_0(\sigma(x))$ you can easily see that $\sigma(x)$ cannot have more than one point other than $0$ (otherwise, you can easily construct a $y\in C(\sigma(x))$ with $0\leq y\leq x$ and $y$ linearly independent with $x$). So $x=\alpha p$ for some projection $p$. The extreme condition requires $p$ to be minimal.

In summary, $A$ has an extreme vector according to that definition if it has a minimal projections. .