Abstract
We prove that the crossed product
C
∗
-algebra
C
∗
r
(Γ,∂Γ)
of a free group
Γ
with its boundary
∂Γ
naturally sits between the reduced group
C
∗
-algebra
C
∗
r
Γ
and its injective envelope
I(
C
∗
r
Γ)
. In other words, we have natural inclusion
C
∗
r
Γ⊂
C
∗
r
(Γ,∂Γ)⊂I(
C
∗
r
Γ)
of
C
∗
-algebras.