Linear maps preserving (p,k) norms of tensor products of matrices

Abstract

Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices. Let $\Vert \cdot {\Vert }_{(p,k)}$ be the $(p,k)$ norm on $M_{mn}$ with $1\le k\le mn$ and $2<p<\infty$. We show that a linear map $\ddot{I}•:M_{mn}\to M_{mn}$ satisfies $$\Vert \ddot{I}•(A\otimes B){\Vert }_{(p,k)}=\Vert A\otimes B{\Vert }_{(p,k)}\mathrm{for}\mathrm{all}A\in M_m\mathrm{and}B\in M_n$$ if and only if there exist unitary matrices $U,V\in M_{mn}$ such that $$\ddot{I}•(A\otimes B)=U(\ddot{I}{†}_1(A)\otimes \ddot{I}{†}_2(B))V\mathrm{for}\mathrm{all}A\in M_m\mathrm{and}B\in M_n,$$ where $\ddot{I}{†}_s$ is the identity map or the transposition map $X\to X^T$ for $s=1,2$. The result is also extended to multipartite systems.