Let $\rho : V_1 \to V_1 $ and $\rho_2 : V_2 \to V_2 $, where $V_1$ and $V_2$ are Hilbert spaces.
Suppose that $U: V_1 \otimes V_2 \to V_1 \otimes V_2$ is a unitary operator.
Define a map $M : L(V_1, V_1) \to L(V_1, V_1)$ as \begin{align*} M(\rho) := \operatorname{Tr}_2 \left(\ U\ \rho\otimes\rho_2 \ U^{\dagger}\ \right) \end{align*} where $\rho_2 \in L(V_2, V_2)$ is a fixed Hermitian, positive-semidefinite operator with $\operatorname{Tr}_2(\rho_2)=1$, and $\operatorname{Tr}_2$ is the partial trace of vector space $V_2$.
Then, trivially $M$ is a unitary operator, if $U = U_1 \otimes U_2$ for some unitary operators $U_1 \in L(V_1,V_1)$ and $U_2 \in L(V_2,V_2)$.
Is the converse also true? If $U$ cannot be expressed as a tensor product of $2$ unitary operators, then is $M$ non-unitary ?
I am lost how to prove this statement. Any hints or references are appreciated. Thank you.