Unique equivariant pseudo expectation and generalized Powers averaging for crossed product

Abstract: Let $Γ$ be a countable discrete group and let $A$ be a unital $Γ$-simple $Γ$-$C<sup>*$-algebra. We prove that $A\rtimes_rΓ$ has the generalized Powers averaging property if and only if the canonical map $A\rtimes_rΓ\to I_Γ(A)$ is the unique $Γ$-equivariant pseudo-expectation. We also show that this averaging property lifts to $I_Γ(A)\rtimes_rΓ$, with the averaging coefficients still belonging to $A$.