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Suppose that $A$ is a $C^*$-algebra and $G$ is a locally compact Hausdorff group acting on $A$.

Gert Pedersen, in his book $C^*$-algebras and their automorphism groups, shows in theorem 7.7.5 that given a faithful representation $\pi$ of a $G$-$C^*$-algebra $A$, the induced regular representation $\tilde \pi \rtimes \lambda$ will be faithful as a representation of $A \rtimes_r A$. However, he does this by showing that the set of states contained in this representation is weak$^*$-dense in the set of states contained the $\tilde \pi_u \rtimes \lambda$, where $\pi_u$ is the universal representation of $A$.

The problem for me is, he defines the reduced crossed product as the image of $\tilde \pi_u \rtimes \lambda$ extended from $C_c(G,A)$ to $A \rtimes A$, and a priori the state argument he then gives is concerning the states contained the representation of $\tilde \pi_u \rtimes \lambda$ considered as a representation of $A \rtimes G$ not $A \rtimes_r G$.

In the above, given a representation $(\pi,H)$ we have for $a \in A$ defined $\tilde \pi(a)\xi(s) := \pi(\alpha_{s^{-1}} (a) ) \xi (s)$, and then for $f \in C_c(G,A)$ and $\xi, \eta \in L^2(G,H)$ we have $$(\tilde \pi \rtimes \lambda (f)\xi, \eta) = \int_G (\tilde \pi(f(s)) \lambda_s \xi , \eta) \mathrm{d} s$$.

I have included relevant screenshots:

screen from book 1

screen from book 2

Can anybody clarify his proof? If needed, I will add more details.

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