Ph.D Thesis | |

Ph.D Student | Salomon Guy |
---|---|

Subject | Operator Algebras of Directed Graphs and Noncommutative Varieties |

Department | Department 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.