Let $A$ be a C*-algebra, $\alpha$ a strongly continuous automorphic action by a locally compact group $G$ on $A$, and consider the crossed product $A\rtimes_\alpha G$. I am looking for references where I can read up on what is known about relations between the state spaces of $A$ and $A\rtimes_\alpha G$. (The case $G={\mathbb R}^n$ would be sufficient for me.)
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In the special case, where $A$ is the group vNa algebra of a type 1 group $H$, and $\alpha$ acts on $H$ via automorphism, then the extremal states on $A \rtimes_\alpha G$ correspond roughly to irreducible representations of $H \rtimes_\alpha G$ (assume that for instance that $H$ is compact), and these can be computed via the Mackey machine (Look at Mackey's paper "Group extensions of locally compact groups"). I would guess that Mark Rieffel has extended this to more general settings, but I am not very knowledgable in this area.
answered May 16, 2012 at 15:00
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$\begingroup$ Thanks for the answer; but I would actually be more interested in this without choosing $A$ in a special way. Is anything general known then? $\endgroup$ Commented May 16, 2012 at 20:00
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$\begingroup$ Here is something about the commutative situation $A=C_0(X)$: folk.uio.no/sergeyn/papers/traces.pdf. I am sorry, I do not have any more information than google can give, or what I have mentioned. So there is probably more than this. $\endgroup$ Commented May 17, 2012 at 8:45