examples of completely positive order zero maps to demonstrate a theorem

Question

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.

Answer 1

Let $K$ be the C*-algebra of compact operators, $A$ a C*-algebra and $\phi:A\to K$ a c.p.c. order zero map.

• $A$ is commutative: assume for simplicity that $A$ is unital. Then there exists a compact Hausdorff space $X$ such that $A\cong C(X)$. In this case $\phi$ is a compression by a positive element $h=\phi(1)\in K$ of a representation of $C(X)$. Observe that the eigenspaces of $h$, which commutes with the representation, are finite-dimensional (I'm excluding the zero from the spectrum, for that simply account for the degeneracy of the representation) and the corresponding projections commute with the representation by functional calculus. This should give you an idea of what $\phi$ looks like.

• $A$ is $M_2,M_3,\ldots,K$: this is essentially the case of the third example referred to in the OP.

• $A$ is simple, non-elementary: consider the CAR algebra as an example. The support $*$-homomorphism $\pi_\phi$ must either be injective or the zero $*$-homomorphism because of the simplicity of $A$. Furthermore $h_\phi=\phi(1)\in K$ commutes with $\pi_\phi$ which implies that there exists a finite rank projection that commutes with $\pi_\phi$. But the CAR algebra has no finite dimensional representations, hence $\phi=0$.