Colloquium

Halmos's dictum, as I learned it, is: “ If you want to study a problem about operators on infinite dimensional Hilbert space, your first obligation is to find formulations or analogues of it in the context of finite dimensional linear algebra. They should help inform your thinking about the infinite dimensional case.” I have spent a considerable amount of time over the years trying to follow it in the context of operator algebras. I will discuss my odyssey in part, describing how it led me to the work of Hochschild in the late 40's, which allows one to describe every finite dimensional algebra in terms of finite graphs, to earlier work of Nevanlinna and Pick on the value-distribution theory of bounded analytic in the open unit disc, to more current work on the structure of Cuntz-Krieger and related C*-algebras. I will focus on examples, leaving technicalities aside. I intend to make the talk accessible to all graduate students.