I hope my question is ok for mathoverflow. I first asked on math.stackexchange but received no answer and then delated it.

I want to understand the proof of the theorem (which you can find in the paper ``Completely positive maps of order zero", by Winter and zacharias, Theorem 2.3 ):

Theorem: Let $A$ and $B$ $C^*-$algebras and $\phi:A\to B$ be a completely postive map of order zero. Set $C:=C^*(\Phi(A))\subset B$.

Then there is a positive element $h\in M(C)\cap C'$ with $\|h\|=\|\phi\|$ and a $*-$homomorphism
$$\pi_{\phi}:A\to M(C)\cap \{h\}'$$ such that $$\phi(a)=\pi_{\phi}(a)h $$ for $a\in A$.

If $A$ is unital, then $\phi(1_A)=h\in C$.

The beginning of the proof: First we assume that $A$ is unital to apply the result for the unital case, as you can find in Wolff's paper about "disjointness preserving operators on $C^*$-algebras". Set $\phi(1_A)=h\in C$. Let $C$ acts nondegenerate on its universal Hilbert space $H$. Wollf's theorem gives us that $h$ lies in the center of $C$. Now it is claimed: (i) h is strictly positive ( this means $\overline{hCh}=C$) and (2) the support projection of $h$ is $1_H$. These claims I want to understand

ad. (i): Let $c_1\in C,\epsilon >0$, why there is $c_2\in C$ such that $\|c_1-hc_2h\|<\epsilon$? I know if you consider $g\in hCh$, it is $g=\phi(1_A)t\phi(1_A)$ with $t\in C$ and t commutes with $\phi(1_A)$. But I'm stuck here, $\phi$ isn't multiplicative.

ad. (ii): What is the exact definition of a support projection here and why is the support projection $1_H$? Maybe the support projection of h can be understood as $\Phi(\chi_{\sigma(h)\setminus\{0\}})=\chi_{\sigma(h)\setminus\{0\}}(h)$, with $\Phi$ is the borel measurable functional calculus of $h$ and $\chi_{\sigma(h)\setminus\{0\}}$ the characteristic function on $\sigma(h)\setminus \{0\}$ and $\chi_{\sigma(h)\setminus\{0\}}(h)$ is an orthogonal projection, which is the identity on $\sigma{(h)}\setminus\{0\}$. But what is the definition/ characteristic property of a support projection in detail and why is it $1_H$? Here it should be important that $C$ acts nondegenerate on $H$.

My third problem is how Stinespring's Theorem is apllied. First they define $$\pi_{\phi}:A\to C^{**}\subseteq B(H),\; a\mapsto s.o. \lim\limits_{n\to\infty}(h+\frac{1}{n}1_H)^{-1}\varphi(a)$$ (s.o.-lim is the strong operator topology limit) which is complitely positve, therefore you can apply Stinespring ( https://en.wikipedia.org/wiki/Stinespring_factorization_theorem ). They say that there is a unital $C^*-$algebra $D$ containing $C^{**}$ and a $*$-homomorphism $$\rho :A\to D$$ such that $$\pi_\phi(a)=1_{C**}\rho(a)1_{C^{**}}$$ for $a\in A$.

My questions: Why does $D$ contain $C^{**}$ (With "$D$ containing $C^{**}$" they mean there is an embedding $i:C^{**}\to D$ which is a nonunital $*-$homomorphism I think)? I think $D$ corresponds to $B(K)$ as stated in Stinespring. Because $A$ and $\pi_\phi$ are unital, $H$ can be considered as a sub Hilbert space of $K$ and it is, up to an unitary, $\pi_{\phi}(a)=P_H\rho(a)_{|H}$ for all $a\in A$. Here is $P_H$ the projection of K onto H. But why is it $\pi_\phi(a)=1_{C**}\rho(a)1_{C^{**}}$ for $a\in A$?

I hope that the topic is specific enough for MO. I appreciate your help. Regards

**Edit**: I found a criterion for (i). For positive elements p in a $C^*$-algebra $A$ it is equivalent:

-$p$ is strictly positive

-Let $u_n=p(\frac{1}{n}+p)^{-1}$, $n\in\mathbb{N}$. It is $(u_n)$ a approximative unit for $A$

h is clearly positive and maybe with this criterion it is easier to prove (i)? I only obtain $\|hu_n-h\|\to 0$ and $\|u_nh-h\|\to 0$, but I need $\|xu_n-x\|\to 0$ and $\|u_nx-x\|\to 0$ for all $x\in A$. Here is $u_n=h(\frac{1}{n}+h)^{-1}$