Boundary maps, germs and quasi-regular representations

@article{Kalantar2022BoundaryMG,
  title={Boundary maps, germs and quasi-regular representations},
  author={Mehrdad Kalantar and Eduardo Scarparo},
  journal={Advances in Mathematics},
  year={2022}
}
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 $\Gamma$-equivariant unital completely positive maps from $\Gamma$-$C^*$-algebras to $C(\partial_F\Gamma)$, where $\partial_F\Gamma$ denotes the Furstenberg boundary of a group $\Gamma$. For a unitary representation $\pi$ coming from the groupoid of germs of a boundary action, we show that there is a unique… 
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