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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.