Groupoid normalisers of tensor products: infinite von Neumann algebras

TL;DR

This paper studies the structure of groupoid normalisers in tensor products of von Neumann algebras, establishing formulas and conditions under which normalisers factorise, with implications for the classification of certain factors.

Contribution

It provides a formula for the groupoid normalisers of tensor product inclusions under specific conditions and characterises when normalisers factorise as tensor products of individual normalisers.

Findings

The normalisers of tensor product inclusions equal the tensor product of individual normalisers under certain conditions.

Normalisers of tensor products factorise as products of individual normalisers when one algebra is finite or both are infinite.

Characterisation of II$_1$ factors with trivial fundamental group for normaliser factorisation.

Abstract

The groupoid normalisers of a unital inclusion $B\subseteq M$ of von Neumann algebras consist of the set $\mathcal{GN}_M(B)$ of partial isometries $v\in M$ with $vBv^*\subseteq B$ and $v^*Bv\subseteq B$. Given two unital inclusions $B_i\subseteq M_i$ of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion $B_1\ \overline{\otimes}\ B_2\subseteq M_1\ \overline{\otimes}\ M_2$ establishing the formula $$ \mathcal{GN}_{M_1\,\overline{\otimes}\,M_2}(B_1\ \overline{\otimes}\ B_2)''=\mathcal{GN}_{M_1}(B_1)''\ \overline{\otimes}\ \mathcal{GN}_{M_2}(B_2)'' $$ when one inclusion has a discrete relative commutant $B_1'\cap M_1$ equal to the centre of $B_1$ (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 $u\in M_1\ \overline{\otimes}\ M_2$ normalising a tensor product $B_1\ \overline{\otimes}\ B_2$ of irreducible subfactors factorises as $w(v_1\otimes v_2)$ (for some unitary $w\in B_1\ \overline{\otimes}\ B_2$ and normalisers $v_i\in\mathcal{N}_{M_i}(B_i)$). 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 II$_1$ factors $B_1$ for which such factorisations always occur (for all $M_1, B_2$ and $M_2$) as those with a trivial fundamental group.