range of trace on projections: beyond rotation algebras

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 $$\tau(K_0(B\rtimes_{(r)} G))\not\subseteq \mu(K_0(B)).$$
• You might want to look at Theorem V.12 in [Rotation numbers for automorphisms of C*-algebras Pacific J. Math., 127 (1987), 31-89]
– Ruy
Oct 31, 2017 at 20:29
• @Ruy I took a quick look and it seems interesting. Thanks for the pointer!
– vap
Oct 31, 2017 at 21:47
• 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. Nov 1, 2017 at 16:35
• @Gabor Szabo It's true there is a certain degree of arbitrariness here. Please see the edit though.
– vap
Nov 2, 2017 at 1:52