Operator algebraic graph theory
The Toeplitz algebra of a directed graph is the norm-closed $*$-algebra generated by edge and vertex concatenation operators on the inner-product space of square summable sequences indexed by finite paths of the graph. A canonical quotient of it is the celebrated Cuntz-Krieger algebra, which is deeply connected to the associated subshift of finite type of the directed graph. Understanding representations of Cuntz-Krieger algebras has become useful for producing wavelet bases on self-similar sets, for encoding properties of slowed continued-fractions expansions and for the development of non-commutative function theory.
In this talk I will present a complete characterization of those finite directed graphs that admit a Cuntz-Krieger representation in which the weak operator topology closed algebra generated by edge and vertex operators (without their adjoints !) is automatically $*$-closed (and hence a von-Neumann algebra). The first example of this counter-intuitive phenomenon was produced by Charles Read in the case where the graph has a single vertex and two loops. I will explain how the Road Coloring theorem of Trahtman (and a periodic version of the theorem due to B\'eal and Perrin) is used to reduce the problem to Read's example.
*Based on joint work with Christopher Linden.