Ex10 28

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 10.28}$

Suppose that subsystem $A = A_1\otimes A_2$ and that $U: A\otimes B \to A\otimes B$ behaves as the identity on $A_1$. In other words, suppose $U = I\otimes V$ where $I$ acts on $A_1$ and $V$ acts on $A_2\otimes B$. Show that for any state $\ket{\phi}$ of system $B$, the superoperator $S_U^\phi$ can be written as $I\otimes S$ for some superoperator $S$ on subsystem $A_2$ alone.