The Radial Masa in a Free Group Factor is Maximal Injective

Details The Radial Masa in a Free Group Factor is Maximal Injective The Radial Masa in a Free Group Factor is Maximal Injective

2008-10-21

arxiv.org/abs/0810.3906v2

J. London Math. Soc. (2), 82 (2010): 787-809

Mathematics

Operator Algebras

25 Pages, Typos corrected and exposition improved

Scientific paper

10.1112/jlms/jdq05

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 also investigate tensor products of maximal injective algebras. Given two inclusions $B_i\subset M_i$ of type $\mathrm{I}$ 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 $B_1\ \bar{\otimes}\ B_2$ is maximal injective in $M_1\ \bar{\otimes}\ M_2$ 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.

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