# Journal of Operator Theory

Volume 69, Issue 2, Spring 2013 pp. 545-570.

Groupoid normalisers of tensor products: infinite von Neumann algebras**Authors**: Junsheng Fang (1), Roger R. Smith (2), and Stuart White (3)

**Author institution:**(1) School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

(2) Department of Mathematics, Texas A \& M University, College Station, Texas 77843, U.S.A.

(3) School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow Q12 8QW, U.K.

**Summary:**The groupoid normalisers of a unital inclusion $B\subseteq M$ of von Neumann algebras consist of the set $\GN_M(B)$ 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 $B_1\ \vnotimes\ B_2\subseteq M_1\ \vnotimes\ M_2$ establishing the formula $$ \GN_{M_1\,\vnotimes\,M_2}(B_1\ \vnotimes\ B_2)''=\GN_{M_1}(B_1)''\ \vnotimes\ \GN_{M_2}(B_2)'' $$ 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 $u\in M_1\ \vnotimes\ M_2$ normalising a tensor product $B_1\ \vnotimes\ B_2$ of irreducible subfactors factorises as $w(v_1\otimes v_2)$ (for some unitary $w\in B_1\ \vnotimes\ B_2$ and normalisers $v_i\in\N_{M_i}(B_i)$). 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.

**Keywords:**normaliser, groupoid normaliser, tensor product, factor, von Neumann algebra

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