Definition of a predual map between predual of von Neumann algebras

Question

Let $M$ and $N$ be two von Neumann algebras. If $T$ is a $*$-isomorphism from $M$ and $N$ , what is the definition of the predual map of $T$? Is it defined as following?

$T_{*}:N_{*}\rightarrow M_{*}$ is the predual map of $T$ such that $T_{*}(\psi)(m)=\psi(T(m))$ for all $\psi \in N_{*}$ and $m \in M$.

Answer 1

Yes. The dual of $T_*$ is $S:M\to N$ given by $$ (Sm)\psi=m(T_*\psi)=(T_*\psi)m=\psi(Tm)=(Tm)\psi. $$ So $Sm=Tm$ for all $m\in M$ and thus $S=T$. That is, $$ (T_*)^*=T $$ so indeed $T_*$ is the predual map of $T$.