Cones and norms in the tensor product of matrix spaces. (Q1426290)

In the tensor product \({\mathbb M}_{n,m}\) of two matrix spaces \({\mathbb M}_m\) and \({\mathbb M}_n\) three natural cones are considered. The first one \({\mathcal P}_0\) consists of positive semi-definite matrices; the second one \({\mathcal P_+}\) is generated by tensor products of positive semi-definite matrices in the factor spaces; \({\mathcal P_-}\) is defined as the dual cone for~\({\mathcal P}_+\). In~\({\mathbb M}_{n,m}\) several tensor norms generated by spectral and/or trace norms in the factor space are considered. The author studies relationships among these norms as well as relations among the three kinds of order intervals in the tensor product.