Boundary operator algebras for free uniform tree lattices

Abstract

Let $X$ be a finite connected graph, each of whose vertices has degree at least three. The fundamental group $Γ$ of $X$ is a free group and acts on the universal covering tree $Δ$ and on its boundary $\partial Δ$, endowed with a natural topology and Borel measure. The crossed product $C^*$-algebra $C(\partial Δ) \rtimes Γ$ depends only on the rank of $Γ$ and is a Cuntz-Krieger algebra whose structure is explicitly determined. The crossed product von Neumann algebra does not possess this rigidity. If $X$ is homogeneous of degree $q+1$ then the von Neumann algebra $L^\infty(\partial Δ)\rtimes Γ$ is the hyperfinite factor of type $III_λ$ where $λ=1/{q^2}$ if $X$ is bipartite, and $λ=1/{q}$ otherwise.