1
$\begingroup$

Let $X$ be an operator system. We denote the set of all unital completely positive maps from $X$ to $M_{n}(\mathbb{C})$ by $UCP_{n}(X)$. How can I characterize $UCP_{n}(M_{n}(\mathbb{C}))$ or $UCP_{n}(C([0,1],\mathbb{C}))$?

For example, is there any relationship between $UCP_{n}(M_{n}(\mathbb{C}))$ and the unitary group of $M_{n}(\mathbb{C})$ or between $UCP_{n}(C([0,1],\mathbb{C}))$ and $M([0,1])$ (i.e. the set of all Borel measures on [0,1])?

$\endgroup$

1 Answer 1

3
$\begingroup$

This is not an answer to your question but a comment.

For c.p. maps either from $M_n(\mathbb{C})$ or into $M_n(\mathbb{C})$, there is a rather simple 1-1 correspondence which may be useful:(you can find the proofs in Brown and Ozawa's exellent book(Proposition 1.5.12 and 1.5.14))

Let A be a C$^*$-algebra and let $ e_{i,j}$ be matrix units of $M_n(\mathbb{C})$. A map $\varphi: M_n(\mathbb{C}) \rightarrow A$ is c.p iff $[\varphi(e_{i,j})]$ is positive in $M_n(A)$.

If A is unital, a map $\phi: A\rightarrow M_n(\mathbb{C})$ is c.p. iff $\phi_n$, the extention of $\phi$ to $M_n(A)$, is positive.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.