Let $M$ be a von Neumann algebra. If $N_1$ and $N_2$ are two subalgebras of $M$ such that $N_1$ is $*$-isomorphic to $N_2$.
What is the relationship between $A_1:=N_1'\cap M$ and $A_2:=N_2'\cap M$? Can we conclude that $A_1$ is $*$-isomorphic to $A_2$?