Groupoid normalizers of tensor products: infinite von Neumann algebras

Fang, J., Smith, R.R. and White, S. (2013) Groupoid normalizers of tensor products: infinite von Neumann algebras. Journal of Operator Theory, 69(2), pp. 545-570.

Abstract

<p>The groupoid normalisers of a unital inclusion B ⊆ M of von Neumann algebras consist of the set GN_M(B) of partial isometries v ∈ M with vBv* ⊆ B and v* Bv ⊆ B. Given two unital inclusions B_i ⊆ M_i of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion B₁ ⊗ B₂ ⊆ M1 ⊗ M2 establishing the formula</p> <p><center>GN_{M1 ⊗ M2} (B₁ ⊗ B₂)" = GN_M1 (B₁)" ⊗ GN_M2 (B₂)"</center></p> <p>when one inclusion has a discrete relative commutant B'₁ ∩ M₁ equal to the centre of B1 b(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 ⊗ M₂ normalising a tensor product B₁ ⊗ B₂ of irreducible subfactors factorises as w(v₁ ⊗ v₂) (for some unitary w ∈ B₁ ⊗ B₂ and normalisers v_i ∈ N_Mi(B_i)). We obtain a positive result when one of the M_i is finite or both of the B_i 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.</p>

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: White, Professor Stuart
• Authors: Fang, J., Smith, R.R., and White, S.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Operator Theory
• ISSN: 0379-4024
• ISSN (Online): 1841-7744