  Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex
matrices. Let $\|\cdot\|_{(p,k)}$ be the $(p,k)$ norm on $M_{mn}$ with $1\leq
k\leq mn$ and $2<p<\infty$. We show that a linear map $\phi:M_{mn}\rightarrow
M_{mn}$ satisfies $$\|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)}
{\rm\quad for~ all\quad}A\in M_m {\rm ~and ~}B\in M_n$$ if and only if there
exist unitary matrices $U,V\in M_{mn}$ such that $$\phi(A\otimes
B)=U(\varphi_1(A)\otimes \varphi_2(B))V {\rm\quad for~ all\quad}A\in M_m {\rm~
and~ }B\in M_n,$$ where $\varphi_s$ is the identity map or the transposition
map $X\to X^T$ for $s=1,2$. The result is also extended to multipartite
systems.
