TITLE: Factor traces of the Temperley-Lieb algebra
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: For $\delta < 2$, I think you can read this off from Jones's "index for subfactors" paper that the "finite depth"-ness forces that the Markov trace is the only trace for each allowed $\delta$.
For $\delta \ge 2$, again from Jones's paper the C$^*$-envelope of $A(\delta)$ is independent of $\delta$, and is in fact a quotient of $C^*(S_\infty)$ (which is easy to see for $\delta=2$ from the Schur-Weyl duality). The factor traces on the latter are classified by the Thoma parameter $0 \le \alpha_n$, $0 \le \beta_n$, $\sum_n \alpha_n + \beta_n \le 1$ ($1 \le n < \infty$), and those induced from $A(\delta)$ correspond to $\alpha_1 + \alpha_2 = 1$ (Wassermann's PhD thesis is the original reference I think?). These correspond to the Markov traces again.
So at the end of the day, the Markov traces are the only factor traces if I'm not mistaken.