Boundaries of Reduced Free Group C*-Algebras

We prove that the crossed product $C^*$-algebra $C^*_r(Γ,\partialΓ)$ of a free group $Γ$ with its boundary $\partialΓ$ naturally sits between the reduced group $C^*$-algebra $C^*_rΓ$ and its injective envelope $I(C^*_rΓ)$. In other words, we have natural inclusion $C^*_rΓ\subset C^*_r(Γ,\partialΓ) \subset I(C^*_rΓ)$ of $C^*$-algebras.

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