# Proof that regular representation induced by faithful representation is faithful on reduced crossed product

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.