Decomposability of extremal positive unital maps on $M_2(\mathbb C)$
A map $\varphi:M_m(\mathbb C)\to M_n(\mathbb C)$ is decomposable if it is of the form $\varphi=\varphi_1+\varphi_2$ where $\varphi_1$ is a CP map while $\varphi_2$ is a co-CP map. It is known that if $m=n=2$ then every positive map is decomposable. Given an extremal unital positive map $\varphi:M_2(\mathbb C)\to M_2(\mathbb C)$ we construct concrete maps (not necessarily unital) $\varphi_1$ and $\varphi_2$ which give a decomposition of $\varphi$. We also show that in most cases this decomposition is unique.