Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras

Abstract

We investigate factoriality, Connes' type III invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type III invariants.Note that the inclusion of continuous cores c(P ) ⊂ c(M ) is the one associated with a fixed faithful normal conditional expectation E P : M → P . The proof of Theorem B generalizes the one of [Io12, Theorem 6.3] and relies on a combination of Popa's deformation/rigidity theory and Connes-Tomita-Takesaki modular theory.Secondly, to understand the position of centralizing nets in M = M 1 * B M 2 , we apply Theorem B to P = M . We want to use Assertion (i) and exploit the key fact that M ∨ B ω (resp. c(M ) ∨ c(B) ω ) can be regarded as an AFP von Neumann algebra. Assuming that M 1 M 1 B and the inclusion B ⊂ M 2 is entirely nontrivial as in Theorem A, we prove that Assertion (iii) in Theorem B cannot occur. On the other hand, we improve Assertion (ii) in Theorem B by showing that c(M 1 ) c(M 1 ) c(B). However, as we already pointed out, this does not contradict our assumption that M 1 M 1 B. Thus, in order to obtain the fullness property for AFP von Neumann algebras, we need to add extra assumptions.When the amalgam B is a type I von Neumann algebra, we obtain the following sharp rigidity result for the asymptotic centralizer M ω , generalizing [Ue12, Theorems 4.8 and 4.10].

Unitary conjugacy for type III subfactors and W$^*$-superrigidity

“…We emphasize that this corollary fails if we do not take tensor products with a type III 1 factor. In fact, there is an inclusion B ⊂ M = A such that M M B but C ϕ (M ) Cϕ(M ) C ϕ (B) (see [HI17,Theorem 4.9]). Hence the type III 1 factor N is necessary.…”