Universal properties of $L(F_\infty)$ in subfactor theory
Sorin Popa and Dimitri Shlyakhtenko
We construct a functor from the category of standard lattices, with embeddings as morphisms, into the category of inclusions $ N\subset M $, with $ N\cong M\cong L(\mathbb {F}_{\infty }) $, with commuting square embeddings as morphisms. This functor is a right inverse to the functor $ \mathcal{G} $ assigning to an inclusion its standard lattice. We also prove that given a subfactor $ Q\subset P $, with $ P $ an arbitrary II$ _{1} $ factor, there exists a subfactor $ \hat{Q}\subset \hat{P} $, with the same standard invariant as $ Q\subset P $, $ \hat{Q}\cong Q*L(\mathbb {F}_{\infty }) $, $ \hat{P}\cong P*L(\mathbb {F}_{\infty }) $.