Passing automorphism group through a representation

Question

This is a very general question-- I'm really after any references to anything similar...

Let $A$ be a $C^*$-algebra equipped with a continuous one-parameter group of automorphisms $(\alpha_t)_{t\in\mathbb R}$ and let $\pi:A\rightarrow B(H)$ be a (non-degenerate) *-representation. I'm after conditions which give a $\sigma$-weakly continuous one-parameter group of automorphisms of $B(H)$, say $(\beta_t)$, such that $\pi\circ\alpha_t = \beta_t\circ\pi$.

As $\beta_t$ will be implemented as $\beta_t(x) = U_t x U_t^*$ where $(U_t)$ is a continuous group of unitaries (that is, a continuous unitary representation of $\mathbb R$). So I'm exactly asking:

When does $\pi$ arise as part of a covariant representation of $(A,\alpha)$?

As covariant representations biject with representations of the crossed product $A \rtimes_\alpha \mathbb R$, it seems that in principle knowing $A \rtimes_\alpha \mathbb R$ should be enough. But this seems terribly abstract!

Answer 1

Borchers, H.-J. On the implementability of automorphism groups. Comm. Math. Phys. 14 1969 305–314. MathSciNet Project Euclid

Borchers defines $\pi$ to be covariant extendible if $\pi$ is (unitarily) equivalent to a subrepresentation of a covariant representation (as it happens, this is enough for my application). Then we have

Theorem: $\pi:A\rightarrow B(H)$ is covariant extendible if and only if for each $\xi,\eta\in H$ the vector state $\mu:a\mapsto (\pi(a)\xi|\eta)$ is such that $\lim_{t\rightarrow 0} \|\mu\circ\alpha_t-\mu\|=0$.

Pedersen's book has lots more, it turns out.