## The radial masa in a free group factor is maximal injective

*Abstract:* The radial (or Laplacian) masa in a free group factor is the abelian von Neumann algebra generated by the sum of the generators (of the free group) and their inverses. The main result of this paper is that the radial masa is a maximal injective von Neumann subalgebra of a free group factor. We establish this result by showing that sequences centralising the radial masa have an orthogonality property in the ultrapower, using a basis introduced by Radulescu. We also investigate tensor products of maximal injective algebras. Given two inclusions B_i\subset M_i of type I von Neumann algebras in finite von Neumann algebras such that each B_i is maximal injective in M_i, we show that the tensor product B_1 \vnotimes B_2 is maximal injective in M_1 \vnotimes M_2 provided at least one of the inclusions satisfies the asymptotic orthogonality property we establish for the radial masa. In particular it follows that finite tensor products of generator and radial masas will be maximal injective in the corresponding tensor product of free group factors.

### Publication Details

*Coauthors:* Jan Cameron, Junsheng Fang and Mohan Ravichandran

J. London. Math. Soc. (2), 82 (2010), 787-809. DOI: 10.1112/jlms/jdq052

*Download:* Journal Website, arXiv:0810.3906, Glasgow ePrint:40625.

The second version on the arXiv and the version on the Glasgow ePrint server reflect the changes made to the paper at page proofs.

## Hammer, Etive Slabs

A surprisingly warm day in Glen Etive on the rucksack club ice climbing meet in February 2008!