Equivalent definition of orthogonal support projections

Question

The above sreentshot is from Takesaki's book.

I have three questions

(1).how to get $\|\varphi_{+}\|=\langle s(\varphi_{+}), \varphi\rangle$?

(2) How to conclude that $\|\varphi_{+}\|=\langle s(\varphi_{+}), \varphi_1\rangle$?

(3)How to verify that $s(\varphi_1)\leq s(\varphi_{+})$?

$s()$ denote by the support projection of a normal positive linear functional

Answer 1

1. The functional $\varphi_+$ is positive, so (since $s(\varphi_+)\leq e$ and so $f\,s(\varphi_+)=0$), $$\|\varphi_+\|=\varphi_+(1)=\varphi_+(s(\varphi_+))=\varphi(e\,s(\varphi_+))=\varphi(e\,s(\varphi_+)-f\,s(\varphi_+))=\varphi(s(\varphi_+)).$$

2. You have that $\|\varphi_+\|\leq\|\varphi_1\|$, $\|\varphi_-\|\leq\|\varphi_2\|$, and $\|\varphi_+\|+\|\varphi_-\|=\|\varphi_1\|+\|\varphi_2\|$.

3. From$\varphi_1(s(\varphi_+))=\|\varphi_1\|$. As $s(\varphi_1)$ is the smallest projection $p$ with $\|\varphi_1\|=\|\varphi(p)\|$, you get that $s(\varphi_1)\leq s(\varphi_+)$.