Let $A \subset B \subset C$ be von Neumann algebras (and more specifically factors, if it helps) either all type II or all type III, acting on the same separable Hilbert space.
Call $A' \subset B$ the "unique complement" of $A$ in $B$, if $A'$ is the least sub-von-Neumann-algebra of $B$ such that the algebra generated by $A \cup A'$ is all of $B$.
Question 1: does $A$ always have a unique complement in $B$?
Question 2: when $A$ has unique complements $A'$ and $A''$, in $B$ and in $C$ respectively (with $B \subset C$), must we have $A' \subseteq A''$?
If anyone can point out examples/proofs I'd be very grateful!