Operator space structures and the split property II

A characterization of the split property for an inclusion $N\subset M$ of $W^*$-factors with separable predual is established in terms of the canonical non-commutative $L^2$ embedding considered in \cite{B1,B2} $$\F_2:a\in N\to \D_{M,\Om}^{1/4}a\Om\in L^2(M,\Om)$$ associated with an arbitrary fixed standard vector $\Om$ for $M$. This characterization follows an analogous characterization related to the canonical non-commutative $L^1$ embedding $$\F_1:a\in N\to (\cdot\Om,J_{M,\Om}a\Om)\in L^1(M,\Om)$$ 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 $\F_i$, $i=1,2$, 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.