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It is a basic fact that every type $II_{1}$ factor posesses a maximal abelian $*$-subalgebra (MASA). My question concerns the concrete realization of such subalgebras. For example, in the free group factor $L\mathbb{F}_{2}$ such a realization is easy, since each of the standard generators generates a MASA.

The Free Burnside group $G=B(2,665)=\langle a,b|g^{665} \rangle$ is infinite, by the work of Adyan and Novikov. Furthermore, the centralizer of any nonidentity element in $G$ is finite cyclic, and so the group is an i.c.c. group and the associated left group von Neumann algebra $LG$ is a type $II_{1}$ factor.

In $LG$, if we want to construct a MASA, it is obvious that the considering the von Neumann subalgebra generated by a given standard generator (and the identity) will not do.

In the free group factor $L\mathbb{F}_{2}$, the self-adjoint element that is the sum of all the standard unitary generators and their inverses generates a maximal abelian *-subalgebra known as the Laplacian MASA. (That this is indeed maximal abelian was originally proved in: T. Pytlik. Radial functions on free groups and a decomposition of the regular representation into irreducible components. J. Reine Angew. Math., 326:124–135, 1981.)

My specific question is this:

If one takes the sum in $LG$ of the standard unitary generators and their inverses, does the resulting element generate a maximal abelian *-subalgebra?

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I haven't thought about this much, but is it easy to see that the algebra it generates is diffuse? –  Owen Sizemore Sep 22 '10 at 22:37
    
I don't even know that, Owen. –  Jon Bannon Sep 22 '10 at 22:49
2  
Good question, but most likely too difficult. 1) Grigorchuk-Zuk showed that the Laplacian of the Lamplighter group has only point spectrum. So it seems to much to hope that the Laplacian algebra is always diffuse. 2) Denis Osin has constructed torsion groups with positive first $\ell^2$-Betti number. Maybe this is a place where one could hope to get that the Laplacian generates a MASA. –  Andreas Thom Sep 23 '10 at 6:06
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