Operator space structures and the split property II

A characterization of the split property for an inclusion $N\subset M$ of $W^{\ast }$-factors with separable predual is established in terms of the canonical non-commutative $L^2$ embedding considered in \cite{B1,B2} $${\text{F}}_2:a\in N\to {\text{D}}_{M,\text{Om}}^{1\mathrm{/}4}a\text{Om}\in L^2(M,\text{Om})$$ associated with an arbitrary fixed standard vector $\text{Om}$ for $M$. This characterization follows an analogous characterization related to the canonical non-commutative $L^1$ embedding $${\text{F}}_1:a\in N\to (\cdot \text{Om},J_{M,\text{Om}}a\text{Om})\in L^1(M,\text{Om})$$ also considered in \cite{B1,B2} and studied in \cite{F}. The split property for a Quantum Field Theory is characterized by equivalent conditions relative to the non-commutative embeddings ${\text{F}}_i$, $i=1,2$, constructed by the modular Hamiltonian of a privileged faithful state such as e.g. the vacuum state. The above characterization would be also useful for theories on a curved space-time where there exists no a-priori privileged state.