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In a rotation algebra, $A_\theta=C(S^1)\rtimes \mathbb{Z}$, there is a tracial state $\tau$ coming from the invariant measure $\mu$ on the circle.

There is a projection $p\in M_n( A_\theta)$ (we can have $n=1$ in this case) such that $\tau(p)=\theta$. In particular we get a value which doesn't come from the commutative part, i.e., $\tau(p)\not\in\mu (K_0(C(S^1)))$.

Are there any other crossed product constructions with the same property?

More precisely: I am asking for an example of a discrete group $G$ and traced unital $C^*$-algebra $(B,\mu)$ such that:

  • $G$ is torsion-free and acts on $(B,\mu)$ in a trace preserving way;
  • letting $\tau$ denote the trace on the crossed product induced by $\mu$, we have $$\tau(K_0(B\rtimes_{(r)} G))\not\subseteq \mu(K_0(B)).$$
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  • $\begingroup$ You might want to look at Theorem V.12 in [Rotation numbers for automorphisms of C*-algebras Pacific J. Math., 127 (1987), 31-89] $\endgroup$
    – Ruy
    Oct 31, 2017 at 20:29
  • $\begingroup$ @Ruy I took a quick look and it seems interesting. Thanks for the pointer! $\endgroup$
    – vap
    Oct 31, 2017 at 21:47
  • $\begingroup$ I have trouble understanding what the question really means. Other crossed product constructions with WHAT property are you after exactly? I could think of several ways how to cherry-pick what "same" could mean in this question. $\endgroup$ Nov 1, 2017 at 16:35
  • $\begingroup$ @Gabor Szabo It's true there is a certain degree of arbitrariness here. Please see the edit though. $\endgroup$
    – vap
    Nov 2, 2017 at 1:52

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