Given $\delta\in\mathbb C$, let $A(\delta)$ denote the complex unital $*$-algebra generated by an identity $1$ and selfadjoint elements $e_k$, $k\in\mathbb N$, satisfying $e_k^2=\delta e_k$, $e_ke_l=e_le_k$ for $|k-l|\geq2$ and $e_ke_{k\pm1}e_k=e_k$ (This puts well-known constraints on the possible values of $\delta$). I wonder if it is known what all the extremal traces (factor traces) of $A(\delta)$ are, i.e. the positive normalized functionals $\tau:A(\delta)\to\mathbb C$ such that $\tau(ab)=\tau(ba)$ for all $a,b\in A(\delta)$ and having the property that $\tau$ cannot be written as a non-trivial convex combination of other traces.

Markov traces are known to be factor traces. But are there additional factor traces on $A(\delta)$, and can they be specified explicitly? Any reference to the literature would be welcome.