Groupoid normalizers of tensor products

We consider an inclusion $B\subseteq M$ of finite von Neumann algebras satisfying $B'\cap M\subseteq B$. A partial isometry $v\in M$ is called a groupoid normalizer if $vBv^*, v^*Bv\subseteq B$. Given two such inclusions $B_i\subseteq M_i$, $i=1,2$, we find approximations to the groupoid normalizers of $B_1 \vnotimes B_2$ in $M_1\vnotimes M_2$, 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 $B_i'\cap M_i\subseteq B_i$, $i=1,2$. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries $v\in M$ satisfying $vBv^*\subseteq B$ and $v^*v, vv^*\in B$.

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