## Groupoid normalizers of tensor products

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

### Publication Details

*Coauthors:* Junsheng Fang, Roger Smith and Alan Wiggins

J. Funct. Anal., 258 (2010), 20-49. DOI: 10.1016/j.jfa.2009.10.005

*Download:* Journal Website, arXiv:0810.0252, Glasgow ePrint:25419.

The arXiv version is the version accepted by JFA. A few typos where corrected at the page proof stage (including the omission of the key hypothesis from the main theorem!). The Glasgow ePrint version corrects these errors, the only differences with the published version should be typesetting choices.

## Stetind, Norway

The national mountain of Norway, an enormous lump of rock coming straight out of the sea, with no path to the top. Summer 2007.