Operator space structures and the split property

A characterization of the split property for an inclusion N⊂M of W∗-factors with separable predual is established in terms of the canonical non-commutative L2 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 L1 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