UNIVERSAL PROPERTIES OF L(F_{∞}) IN SUBFACTOR THEORY

SORIN POPA DIMITRI SHLYAKHTENKO

Abstract. We construct a functor from the category of standard lattices, with embeddings as morphisms, into the category of inclusions N ⊂ M, with N ≅ M ≅ L(𝔽_{∞}), with commuting square embeddings as morphisms. This functor is a right inverse to the functor G assigning to an inclusion its standard lattice. We also prove that given a subfactor Q ⊂ P, with P an arbitrary II_{1} factor, there exists a subfactor Q̂ ⊂ P̂, with the same standard invariant as Q ⊂ P, Q̂ ≅ Q ∗ L(𝔽_{∞}), P̂ ≅ P ∗ L(𝔽_{∞}).