Reference for the G-equivariant Stinespring dilation theorem

Question

Is there a good reference for the G-equivariant Stinespring dilation theorem? I can't find the theorem stated anywhere.

Thanks in advance.

Answer 1

I found it: Lemma 3.1 in "Equivariant Completely Bounded Operators" by Raeburn, Sinclair, and Williams.

Please seek out the paper before relying on what I copied here

For completeness it is stated as:

Definitions: Let $\alpha :G\to$ Aut $A$ and $\beta:G\to$ Aut $B$ be actions of an amenable locally compact group $G$ on $C^*$-algebras A and B.

Let $\Phi:(A,G,\alpha)\to (\mathscr{B}(\mathscr{H}),G,Ad U)$ be an equivariant completely positive map. Then there are

1. a covariant representation $(\pi, W)$ of $(A, G, \alpha)$ on a space $\mathscr{K}$,
2. a continuous operator $V:\mathscr{H}\to\mathscr{K}$, and
3. a representation $\rho$ of the commutant $\Phi(A)'$ on $\mathscr{K}$,

such that

1. $\Phi(a) = V^* \pi(a) V$ for all $a \in A$,
2. $\mathscr{K} = (\pi(A)V\mathscr{H})^-$,
3. $VU_t = W_t V$ for all $t \in G$, and
4. $\rho(x) V = Vx$ for $x \in \Phi(A)', \rho(U_t x U_t^*) = W_t \rho(x) W_t^*$ for $x \in \Phi(A)', t \in G,$ and $\rho(\Phi(A)') \subseteq \pi(A)'$.