I'm interested explicit examples which can be used to demonstate the theorem:

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 $\phi(1_A)=h\in C$.

It can be found in the paper "Completely positive maps of order zero", by Winter and zacharias, theorem 2.3.

Three examples are: A trivial one, if $A$ and $B$ are unital and $\phi:A\to B$ is a unital $\ast$-homomorphism. Then nothing exciting happens, it is $\phi=\pi_{\phi}$.

A second one, for $g\in C([0,1])$ a positive function, $\phi_g:C([0,1])\to C([0,1]),\; f\mapsto gf$, it is $\phi_g(f)=g\pi_{\phi}(f)$ with a $\ast$-homomorphism $\pi_{\phi}:C([0,1])\to C_b(\{x\in [0,1]: g(x)\neq 0\}),\; f\mapsto f$ (here I was inspired by an other question on MSE).

A third one, which can be found here https://math.stackexchange.com/questions/1171402/order-zero-maps-in-matrix-algebra .

My question is: do you know an other example (a nontrivial one) (something with compact operators on a Hilbert space or integral oeprators for example)? Regards.