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.

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Publisher's URL: http://www.mathjournals.org/jot/2013-069-002/index.html

Abstract

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 ⊗ B2 ⊆ M1 ⊗ M2 establishing the formula

GNM1 ⊗ M2 (B1 ⊗ B2)" = GNM1 (B1)" ⊗ GNM2 (B2)"

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 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.

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

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