Let $A$ be a (unital) $C^*$-algebra and let $\alpha: G\to Aut(A)$ be an action of $G$ on $A$, where $G$ is a discrete (countable) group.
Denote by $\pi: A \rtimes_{\operatorname{\alpha}}G \to A \rtimes_{\operatorname{\alpha,r}}G$ the canonical quotient map from the full crossed product onto the reduced.
Denote by $E:A \rtimes_{\operatorname{\alpha,r}}G \to A$ the faithful conditional expectation onto $A$ satisfying $E(\sum a_gg)=a_e$, where $\sum a_gg \in AG$ (or $C_c(G,A)$) is finitely supported function.
I have seen in lecture notes as a fact the following argument:
$ker(\pi)= \{x\in A \rtimes_{\operatorname{\alpha}}G: E(xu^s)=0 , \forall s\in G\}$

I don't know how to explain it. I know that we have uniqueness of coefficients in $A \rtimes_{\operatorname{\alpha,r}}G$ by faithfulness of $E$, and in the above I think they mean $E \circ \pi$. probably I need to combine these facts in order to get the result, but I'm not sure how...

Thank you!


This is not properly an answer, due to lack of knowledge and a fear of cross products.

What I can contribute, though, is that it should clearly be $E\circ \pi$. The other observation is that, assuming that all elements of $A\rtimes_{\alpha,r}G$ are of the form $y=\sum_s a_su^s$ (maybe not in norm? At least in the case of $C^*_r(G)$ they are $\|\cdot\|_2$-convergent), then $$ a_s=E(yu^{-s}).$$Maybe this was already obvious to you. In any case, I'm not sure if I can make this work formally.


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