Boundary maps, germs and quasi-regular representations

Abstract

We investigate the tracial and ideal structures of C⁎-algebras of quasi-regular representations of stabilizers of boundary actions.Our main tool is the notion of boundary maps, namely Γ-equivariant unital completely positive maps from Γ-C⁎-algebras to C(∂FΓ), where ∂FΓ denotes the Furstenberg boundary of a group Γ.For a unitary representation π coming from the groupoid of germs of a boundary action, we show that there is a unique boundary map on Cπ⁎(Γ). Consequently, we completely describe the tracial structure of the C⁎-algebras Cπ⁎(Γ), and for any Γ-boundary X, we completely characterize the simplicity of the C⁎-algebras generated by the quasi-regular representations λΓ/Γx associated to stabilizer subgroups Γx for any x∈X.As an application, we show that the C⁎-algebra generated by the quasi-regular representation λT/F associated to Thompson's groups F≤T does not admit traces and is simple.