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 $$\tau(K_0(B\rtimes_{(r)} G))\not\subseteq \mu(K_0(B)).$$