Ph.D Thesis

Ph.D StudentSalomon Guy
SubjectOperator Algebras of Directed Graphs and Noncommutative
DepartmentDepartment of Mathematics
Supervisor ASSOCIATE PROF. Orr Moshe Shalit


A (concrete) operator algebra on a Hilbert space H is a closed subalgebra of the algebra B(H) of bounded operators on a Hilbert space H. Many operator algebras arise in a natural way from other types of mathematical objects. This thesis focuses on two such examples. Both have a geometric flavor: operator algebras that arise from directed graphs and operator algebras that arise from noncommutative varieties.

In the first part of this thesis, we consider operator algebras arising from directed graphs. A directed graph can be thought of as a countable set of vertices with directed arrows (known as edges) between them. Each directed graph gives rise to an operator algebra generated by a set of projections indexed by the graph vertices and a set of partial isometries indexed by the graph edges. The relations these generators satisfy fully encode the geometry of the graph.

This operator algebra sits canonically inside two well-known operator algebras that are self-adjoint (such operator algebras are called C*-algebras), and, in fact, it generates them. In this situation, where a C*-algebra contains a generating operator algebra, a question that naturally arises is how rigid this containment is. There are several relevant notions of rigidity that may be considered, and here we try to address some of them. More specifically, we study those representations of the graph operator algebra that have unique extensions (in some sense) to the C*-algebra containing it. In this way, we understand a rather strong type of rigidity, called hyperrigidity. We also shed light on a rather weak type of rigidity, studied before, that is expressed in the fact that one of the above C*-algebras is the so-called C*-envelope of the graph operator algebra. Our results also give rise to some new generalizations of classical results in dilation theory.

In the second part, we turn our attention to noncommutative (nc, for short) varieties. Here, the playground is the d-dimensional nc ball, that is, the disjoint union of all d-dimensional matrix unit balls, and the players are bounded nc functions on it, namely, bounded limits of free polynomials in d variables. An nc variety is then simply a subvariety of the nc ball cut out by a set of bounded nc functions. The operator algebra that we associate to an nc variety is the algebra of bounded nc functions on the variety. Here we address a somewhat more basic question (that has been solved for the operator algebras associated to directed graphs): when are two such algebras isomorphic in terms of the underlying varieties?

We try to answer this question in several categories; in some we obtain full characterizations, while in others we have to restrict ourselves to a smaller class of varieties. Along the way we study some related properties and present some noncommutative counterparts of classical properties in multidimensional complex analysis, such as Hilbert Nullstellensatz, the maximum modulus principle, and the Schwarz lemma of holomorphic functions on the disc.