## Decomposability and 2-positivity of linear map from $M_3(\mathbb{C})$ to $M_9(\mathbb{C})$

publié le

Gunjan Sapra
(Kyoto university)

We give a criterion which is a necessary and sufficient condition for a linear map to be k-positive and show that how equivariance is a useful to conclude about k-positivity of linear maps on matrix algebras. We define a family of linear maps from $M_3(\mathbb{C})$ to $M_9(\mathbb{C})$ and study the properties of positivity, completely positivity, 2-positivity and decomposability. It has been proved that every 2-positive linear map from $M_3(\mathbb{C})$ to $M_3(\mathbb{C})$ is decomposable. It would be interesting to find whether decomposability of a linear map implies its 2-positivity. We give a partial answer to this question in the case of linear maps from $M_3(\mathbb{C})$ to $M_9(\mathbb{C})$.