Abstract

We consider the Thompson group $F$ given by its infinite presentation$$F = \big\langle x_0, x_1, x_2, \ldots \mid \ x_jx_i=x_ix_{j+1}\ {\rm for } \ j>i\big\rangle.$$ and study the Toeplitz C-algebras ${\cal T}\lambda(M_k)$ of the submonoids $P_k$ of $F$ generated by the first $k+1$ generators. We obtain a spanning set, a characterization of faithful representations, and we show that the boundary quotients are purely infinite simple. ${\cal T}\lambda(M_0)$ is the C-algebra of the unilateral shift,${\cal T}\lambda(M_1) \cong {\cal TO}_2$ is known to be nuclear by results of Cuntz; and nuclearity of ${\cal T}\lambda(M_2)$ follows from work of an Huef, Nucinkis, Sehnem and Yang. We prove nuclearity of ${\cal T}\lambda(M_k)$ for $k =3, 4$ and discuss our approach to the general case using a realization of ${\cal T}\lambda(M_k)$ and its boundary quotient $\partial {\cal T}_\lambda(M_k)$  as the Toeplitz algebra and the Cuntz—Pimsner algebra of a C*-correspondence.  This is joint work with A. an Huef, B. Nucinkis, I. Raeburn and C. Sehnem.