Given a C*-algebra $\mathcal{A}$.
Consider a state: $\omega\geq0$
Especially one has: $\sup\omega(E^2)=\|\omega\|$
Can it actually fail to be a proper limit?
The problem is that the square is not operator-monotonic: $$A,B\geq0:\quad A\leq B\nRightarrow A^2\leq B^2$$ But that seems crucial when extracting from the enclosing inequality: $$|\omega(A)|\leftarrow|\omega(AE)|\leq\omega(A^*A)\omega(E^2)\leq\|\omega\|_+\omega(E^2)\leq\|\omega\|_+^2\leq\|\omega\|^2$$ Does somebody have an alternative or a counterexample?
Instead, the norm-identity follows from: $\lim\omega(E)=\sup\omega(E)=\|\omega\|_+=\|\omega\|$
(Besides, Bratelli & Robinson seem to have a tiny mistake here in their proof.)