I have a question about a passage in the proof (on page 316) of proposition 3.2 in this paper http://wwwmath.uni-muenster.de/42/fileadmin/Einrichtungen/mjm/vol_2/mjm_vol_2_14.pdf. My question is related to the uniqueness of the extension.
Let $\varphi:A\to B$ a completely positive order zero map, $\varphi_1:A_1\to C^{\ast\ast}$ and $\psi:A_1\to C^{\ast\ast}$ extensions of $\varphi$ with $\varphi_1(1_{A_1})=g$, $\psi(1_{A_1})=d$ and $s.o.\lim\limits_{\lambda}\psi(u_\lambda)=g$. Clearly, $d\ge g$.
Suppose that $\|d-g\|>0$, why there are $\eta>0$ and $\lambda\in\Lambda$ such that $\|(d-g)\varphi(u_\lambda)(d-g)\|\ge \eta$?
The first and last passage which I don't understand is the next one:
Using functional calculus, one finds positive Elements $u,w\in A$ of norm at most one such that $\|u-u_\lambda\|<\frac{\eta}{2}$ and $wu=u$. I don't know how to prove it. Maybe I have to use the continuous function $f_n(x)=x(\frac{1}{n}+x)^{-1}$ on $[0,\infty)$ once again, but I don't know.
Could you help me? The rest of the proof is clear, I only stuck on this passage. Regards