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Decomposability of extremal positive unital maps on ${M}_{2}\left(ℂ\right)$. (English) Zbl 1116.47033
Bożejko, Marek (ed.) et al., Quantum probability. Papers presented at the 25th QP conference on quantum probability and related topics, Będlewo, Poland, June 20–26, 2004. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 73, 347-356 (2006).

Let ${M}_{n}\left(ℂ\right)$ be the algebra of matrices with complex entries. For $m,n$ in $ℕ$, we have

${M}_{m}\left({M}_{n}\left(ℂ\right)\right)\cong {M}_{m}\left(ℂ\right)\otimes {M}_{n}\left(ℂ\right)\cong {M}_{mn}\left(ℂ\right)·$

Thus ${M}_{m}\left({𝕄}_{n}\left(ℂ\right)\right)$ has the structure of a ${C}^{*}$-algebra. A matrix $A\in {M}_{m}\left({M}_{n}\left(ℂ\right)\right)$ is positive iff

$\sum _{i,j=1}^{m}\overline{{\mu }_{i}}{\mu }_{j}<{\vartheta }_{i},\phantom{\rule{4pt}{0ex}}{A}_{ij}{\vartheta }_{j}\le 0\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{every}\phantom{\rule{4.pt}{0ex}}{\vartheta }_{1},\cdots ,{\vartheta }_{m}\in {ℂ}^{n}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{\mu }_{1},\cdots ,{\mu }_{m}\in ℂ·$

A linear map $\varphi :{M}_{m}\left(ℂ\right)\to {M}_{n}\left(ℂ\right)$ is called positive if $\varphi \left(A\right)$ is a positive matrix for every positive matrix $A\in {M}_{m}\left(ℂ\right)$. If $k\in ℕ$, $\varphi$ is called $k$-positive map (resp., $k$-copositive map) whenever $\left[\varphi {\left(A\right)}_{ij}\right){\right]}_{i,j=1}^{k}$ (resp., $\left[\varphi {\left(A\right)}_{j,i}\right){\right]}_{i,j=1}^{k}\right)$ is a positive element in ${M}_{k}\left({M}_{n}\left(ℂ\right)\right)$ for every positive ${\left[{A}_{ij}\right]}_{i,j=1}^{k}$ in ${M}_{k}\left({M}_{m}\left(ℂ\right)\right)$. If $\varphi$ is $k$-positive (resp., $k$-copositive) for every $k\in ℕ$, then $\varphi$ is called completely positive (resp., completely copositive).

A positive map which is a sum of completely positive and completely copositive maps is called decomposable. It is known that if $m=n=2$, then every positive map is decomposable. Given an extremal unital positive map $\varphi :{M}_{2}\left(ℂ\right)\to {M}_{2}\left(ℂ\right)$, the authors construct concrete maps ${\varphi }_{1}$ and ${\varphi }_{2}$ which give a decomposition of $\varphi$. They also show that in most cases this decomposition is unique.

##### MSC:
 47B65 Positive and order bounded operators 47L07 Convex sets and cones of operators
##### Keywords:
positive maps; decomposable maps; face structure