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.

DOI: http://dx.doi.org/10.7900/jot.2011mar05.1928
Keywords: normaliser, groupoid normaliser, tensor product, factor, von Neumann algebra

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