Tracial states on $C_\text{r}^\ast(\mathbf{D}_\infty)$, $\mathbf{D}_\infty$ being the infinite dihedral group

Question

Let $\mathbf{D}_\infty$ be the infinite dihedral group, i.e. the group generated by two elements $s$ and $t$ with $s^2=t^2 =e$ which are free with respect to each other. Consider the reduced group $C^\ast$-algebra $C_\text{r}^\ast (\mathbf{D}_\infty)$. There exist two canonical tracial states on this $C^\ast$-algebra, namely the state $\tau$ with $\tau(1)=1$ and $\tau(\lambda_s)=\tau(\lambda_t)=0$ and (since $\mathbf{D}_\infty$ is amenable) the character coming from the trivial representation of the group $\mathbf{D}_\infty$ (i.e. the corresponding state maps $1\mapsto1$, $\lambda_s\mapsto1$, $\lambda_t \mapsto 1$). Of course convex combinations of these two states are also tracial, but are these all the tracial states? If no, is there a complete description of all tracial states on $C_\text{r}^\ast(\mathbf{D}_\infty)$? How do they look like?

Answer 1

G. K. Pedersen proved that $C^*(D_\infty )$ is isomorphic to the subalgebra of $C([-1, 1], M_2)$ formed by all continuous functions $$ f:[-1,1]\to M_2, $$ such that $f(-1)$ and $f(1)$ are diagonal, via an isomorphism taking the two generators $s$ and $t$ to the functions $f_s$ and $f_t$ defined by $$ f_s(x) = \pmatrix{ 1 & 0 \cr 0 & -1}, \quad \text {and} \quad f_t(x) = \pmatrix {x & \sqrt{1-x^2} \cr \sqrt{1-x^2} & -x}. $$

If $\mu $ is any probability measure on $[-1, 1]$ then the functional $$ \tau (f) = \frac 12\int_{-1}^1 \text{tr}(f(x))\, d\mu (x) $$ is a trace. There are four other traces on this algebra, actually characters, not covered by the above class of examples, given by $$ \tau ^-_{1}(f) = f(-1)_{1, 1}, $$ $$ \tau ^-_{2}(f) = f(-1)_{2, 2}, $$ $$ \tau ^+_{1}(f) = f(1)_{1, 1}, $$ $$ \tau ^+_{2}(f) = f(1)_{2, 2}, $$ so there are many more traces than the ones described by the OP. On the other hand, I believe the above span all traces of $C^*(D_\infty )$.