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:



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



Passing automorphism group through a representation

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> Borchers, H.-J.
>  On the implementability of automorphism groups.
>  Comm. Math. Phys. 14 1969 305–314.
>  [MathSciNet](http://www.ams.org/mathscinet-getitem?mr=259631) [Project Euclid](http://projecteuclid.org/euclid.cmp/1103841819)
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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



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> **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 $\lim\_{t\rightarrow 0} \|\mu\circ\alpha\_t-\mu\|=0$.
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Pedersen's book has lots more, it turns out.


