Groupoid normalisers of tensor products: infinite von Neumann algebras

The groupoid normalisers of a unital inclusion B ⊆ M of von Neumann algebras consist of the set gNM (B) of partial isometries v ∈ M with vBv* ⊆ B and v* Bv ⊆ B. Given two unital inclusions Bi ⊆ Mi of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion B1 {position indicator} B2 ⊆ M1 {position indicator} M2 establishing the formula, when one inclusion has a discrete relative commutant B′1 ∩ M1 equal to the centre of B1 (no assumption is made on the second inclusion). This result also holds when one inclusion is a generator masa in a free group factor. We also examine when a unitary u ∈ M1 {position indicator} M2 normalising a tensor product B1 {position indicator} B2 of irreducible subfactors factorises as ω(v1 ⊗ v2) (for some unitary ω ∈ B1 B2 and normalisers vi ∈ NMi (Bi)). We obtain a positive result when one of the Mi is finite or both of the Bi are infinite. For the remaining case, we characterise the II1 factors B1 for which such factorisations always occur (for all M1, B2 and M2) as those with a trivial fundamental group. © Theta, 2013.

Groupoid normalisers of tensor products: infinite von Neumann algebras Academic Article