Boundary Operator Algebras for Free Uniform Tree Lattices

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 ∆ and on its boundary ∂∆, endowed with a natural topology and Borel measure. The crossed product C∗-algebra C(∂∆)⋊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∞(∂∆)⋊Γ is the hyperfinite factor of type IIIλ where λ = 1/q 2 if X is bipartite, and λ = 1/q otherwise.