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Tuesday, June 12, 2018

**Abstract:** 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. In the theory of free semigroup algebras, this is done for Toeplitz-Cuntz algebras, and is achieved via two key results in the theory: the first is a theorem of Davidson, Katsoulis and Pitts on the $2\times 2$ structure of free semigroup algebras, and the second is a Lebesgue-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 theorem. We discuss some of the difficulties and similarities when passing to the more general context of operator algebras associated to directed graphs.