A simple proof of the Jamiolkowski criterion for complete positivity of   linear maps of algebras of Hilbert-Schmidt operators
  We generalize a preceding simple proof of the Jamiolkowski criterion to check whether a given linear map between algebras of operators is completely positive or not. The generalization is performed to embrace all algebras of Hilbert-Schmidt class operators, thus possibly infinite-dimensional. 
D. Salgado and J.L. Sanchez-Gomez