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In the paper "Orbit Equivalence and Topological Conjugacy of Affine Actions on Compact Abelian Groups", S. Bhattacharya shows (Theorem 3) the following:

Theorem. Given two actions $\alpha$ and $\beta$ of a discrete group $\Gamma$ on a compact connected metrizable abelian group $K$ by continuous group automorphisms the following are equivalent:

  1. there exists a homeomorphism $F\colon K \to K$ such that $\alpha_\gamma= F\beta_\gamma F^{-1}$ for every $\gamma\in \Gamma$.

  2. there exists a continuous group automorphism $F\colon K \to K$ such that $\alpha_\gamma = F\beta_\gamma F^{-1}$ for every $\gamma \in \Gamma$.

By passing to the Pontryagin dual this becomes a statement about discrete countable torsion-free abelian groups. My question concerns the generalization to non-abelian groups.

Question. Given two actions $\alpha$ and $\beta$ of a discrete group $\Gamma$ on a discrete countable torsion-free group $G$ by group automorphisms, are the following equivalent?

  1. there exists an automorphism of the reduced $C^*$-algebra of $G$ which conjugates the actions induced by $\alpha$ and $\beta$

  2. there exists an automorphism of $G$ which conjugates $\alpha$ and $\beta$.

This question is also interesting when we restrict the attention fo $\Gamma = \mathbb Z$ (i.e. to pairs of automorphisms.)

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Forgot an exponent "2" in the last line? –  André Henriques May 12 '11 at 17:00
@André: You mean it should be $\mathbb Z^2$? Perhaps it is easy for Z, but I don't see it. $\Gamma = \mathbb Z$ means that we have a pair of automorphisms (each automorphism induces an action of $\mathbb Z$) and I ask whether "topological" conjugacy of these automorphisms implies "algebraic" conjugacy. –  Łukasz Grabowski May 13 '11 at 15:03
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