Scaling limit for subsystems and Doplicher-Roberts reconstruction

Given an inclusion $B \subset F$ of (graded) local nets, we analyse the structure of the corresponding inclusion of scaling limit nets $B_0 \subset F_0$, giving conditions, fulfilled in free field theory, under which the unicity of the scaling limit of $F$ implies that of the scaling limit of $B$. As a byproduct, we compute explicitly the (unique) scaling limit of the fixpoint nets of scalar free field theories. In the particular case of an inclusion $A \subset B$ of local nets with the same canonical field net $F$, we find sufficient conditions which entail the equality of the canonical field nets of $A_0$ and $B_0$.