You can see the proposition in a google books preview here. First and foremost, my question is:
Question: Am I correct to interpret Proposition 2.39 as setting up a bijective correspondence between
- The set of nondegenerate covariant homomorphisms $(\pi,u) : (A,G,\alpha) \to \mathcal{L}(X)$.
- The set of nondegenerate homomorphisms $L : A \rtimes_\alpha G \to \mathcal{L}(X)$.
by sending each $(\pi,u)$ to $L = \pi \rtimes u$ and sending each $L$ to the $(\pi,u)$ defined by $\pi(a) = \overline{L}(i_A(a)), u_s = \overline{L}(i_G(s))$ for all $a \in A, s \in G$.
I don't really have a mathematical reason to doubt this reading is correct. My reasons are slightly meta, so I hope I am making myself understood. The immediately following Proposition 2.40 would seem, in part, to be a corollary, where the Hilbert $B$-module $X$ is taken to be a Hilbert space.
Now, in Proposition 2.40 Williams states explicitly that a bijective correspondence is being set up, whereas in Proposition 2.39 this is only implicit. That in and of itself would not be enough to raise my eyebrows, but, have a look at the 1st paragraph of the proof of Proposition 2.40.
"Proposition 2.39 on the facing page shows that the map $(\pi,U) \mapsto \pi \rtimes U$ is a surjection. It's one-to-one in view of Equations (2.21) and (2.27)."
I don't understand the need for the reference to equations (2.21) and (2.27). Does Proposition 2.40 not already show we have a bijection?