Event № 816

Representations of Toeplitz-Cuntz algebras were studied by Davdison, Katsoulis and Pitts via non-self-adjoint techniques, originating from work of Popescu on his non-commutative disk algebra. This is accomplished by working with the WOT closed algebra generated by operators corresponding to vertices and edges in the representation. These algebras are called free semigroup algebras, and provide non-self-adjoint invariants for representations of Toeplitz-Cuntz algebras.

The classication of Cuntz-Krieger representations of directed graphs up to unitary equivalence was used in producing wavelets on Cantor sets by Marcolli and Paolucci and in the study of semi-branching function systems by Bezuglyi and Jorgensen. With Davidson and B. Li we extended the theory of free semigroup algebras to arbitrary directed graphs, where free semigroupoid algebras provide new connections with graph theory.

In this talk I will present a characterization of those finite directed graphs that admit self-adjoint free semigroupoid algebras. We will make full circle with the theory of automata, as we will use a periodic version of the Road Coloring theorem due to Beal and Perrin, originally proved by Trahtman in the aperiodic case. This is based on joint work with Christopher Linden.