Groupoid normalizers of tensor products

TL;DR

This paper characterizes the structure of groupoid normalizers in tensor products of finite von Neumann algebra inclusions, showing they generate the tensor product of the individual normalizer algebras under certain conditions.

Contribution

It provides a new approximation method for groupoid normalizers in tensor products and establishes their algebraic structure, extending prior results with specific hypotheses.

Findings

The von Neumann algebra generated by groupoid normalizers of the tensor product equals the tensor product of their individual algebras.

Approximation techniques for groupoid normalizers in tensor products are developed.

Examples demonstrate the necessity of the hypothesis for the main result to hold.

Abstract

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 $v^*v, vv^*\in B$.