By a result of Glimm, we know that classifying representations of non-type-I $C^*$-algebras up to unitary equivalence is essentially impossible (at least with countable structures). Instead of this, one either restricts to a tractable subclass or weakens the invariant.

Using the theory of free semigroup algebras, the latter is achieved for Toeplitz-Cuntz algebras. Two key results in this theory enable this: the first is a theorem of Davidson, Katsoulis and Pitts on the $2\times 2$ structure of free semigroup algebras, and the second, a Lebesuge-von Neumann-Wold decomposition theorem of Kennedy.

This talk is about joint work with Ken Davidson and Boyu Li, where we generalize this theory to representations of Toeplitz-Cuntz-Krieger algebras associated to a directed graph $G$. We prove a structure theorem akin to that of Davidson, Katsoulis and Pitts, and provide a Lebesuge-von Neumann Wold decomposition using Kennedy’s results. We discuss some of the difficulties and similarities when passing to the more general context of operator algebras associated to directed graphs.