Faithful extreme traces on group C*-algebras

Question

Let $G$ be a discrete amenable, residually finite, ICC(i.e. each non-trivial conjugacy class is infinite) group. Let $C^*_r(G)$ be the reduced group $C^*$-algebra of $G$. Since $G$ is ICC the (faithful) canonical trace $\tau$ that maps every non-trivial group element to 0 is an extreme trace.

Is there a group $G$, satisfying the above properties and an extreme trace $\tau’\neq\tau$ on $G$ that is faithful on $C^*_r(G)$, i.e. $\tau’(x^*x)>0$ for all non-zero $x\in C^*_r(G)$?

I would be interested in an example where any of the adjectives amenable,RF or ICC (in this case we just want two distinct extreme faithful traces) are dropped but my main interest is when all three adjectives are present.

Answer 1

The lamplighter group $G = (\mathbb{Z}/2\mathbb{Z}) \wr \mathbb{Z}$ is such an example. The group is amenable, ICC and residually finite. The C$^*$-algebra $C^*_r(G)$ can be identified with the crossed product of $\mathbb{Z}$ acting by the shift on the Cantor space $X = \{0,1\}^{\mathbb{Z}}$. For every $t \in (0,1)$, we have the $\mathbb{Z}$-invariant probability measure $\mu_t$ on $X$ given by the infinite product of the probability measure on $\{0,1\}$ that assigns measure $t$ to $\{0\}$. The measure $\mu_t$ uniquely extends to a trace on $C^*_r(G)$ that is extremal and faithful.