Maximal amenable MASAs of radial type in the free group factors

Abstract

We prove that if $\{(M_j,{\tau }_j){\}}_{j\in J}$ are tracial von Neumann algebras, $s_j\in M_j$ are selfadjoint semicircular elements and $t=(t_j{)}_j$ is a square summable $J$-tuple of real numbers with at least two non-zero entries, then the von Neumann algebra $A(t)$ generated by the ``weighted radial element'' ${\sum }_j{t}_j{s}_j\in M:={\ast }_{j\in J}{M}_j$ is maximal amenable in $M$, with $A(t)$, $A(t^{\prime })$ unitary conjugate in $M$ iff $t,t^{\prime }$ are proportional. Letting $M_j$ be diffuse amenable, $\forall j$, this provides a large family of maximal amenable MASAs in the free group factor $L{\mathbb{F}}_n$.

arXiv comment6 pages, paper to appear in Proc. AMS