Lattices of intermediate subfactors for type III factors

Abstract.

Let M be a factor and N a subfactor of M with finite index. If N is an irreducible subfactor of M, then the intermediate subfactor lattice for the inclusion $N \subset M$ is a finite lattice. In a common discrete decomposition intermediate subfactor lattices of type II and type III inclusions are different in general. But they are isomorphic in a common continuous decomposition of a type III₁ inclusion. We determine the structure of lattices of subfactors with index 4 without the assumption of AFD.