Groupoid normalizers of tensor products

We consider an inclusion B ⊆ M of finite von Neumann algebras satisfying B ∩ M ⊆ B. A partial isometry v ∈M is called a groupoid normalizer if vBv∗, v∗Bv ⊆ B. Given two such inclusions Bi ⊆Mi , i = 1, 2, we find approximations to the groupoid normalizers of B1 ⊗ B2 in M1 ⊗ M2, from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis B i ∩Mi ⊆ Bi , i = 1, 2.We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries v ∈M satisfying vBv∗ ⊆ B and v∗v, vv∗ ∈ B. © 2009 Elsevier Inc. All rights reserved.