## Groupoid normalisers of tensor products: infinite von Neumann algebras

*Abstract:* 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\ \overline{\otimes}\ B_2\subseteq M_1\ \overline{\otimes}\ M_2 establishing the formula GN_{M_1\,\overline{\otimes}\,M_2}(B_1\ \overline{\otimes}\ B_2)''=GN_{M_1}(B_1)''\ \overline{\otimes}\ 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 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.

### Publication Details

*Coauthors:* Junsheng Fang and Roger Smith.

J. Oper. Thy., 69(2), 545-570, 2013. DOI: 10.7900/jot.2011mar05.1928

*Download:* Journal Website, arXiv:1004.0851, Glasgow ePrint 58640/

The arXiv version is the version accepted by JOT.

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Easter 2007.