Much of my work in Functional Analysis involves C*-algebras constructed from discrete and dynamical structures. This involves building C*-algebras from objects (e.g., graphs, matrices, bimodules, shift spaces) and then using the object to study properties of the associated C*-algebra. In these investigations one often considers the following questions: When can a known C*-algebra be modeled as the C*-algebra of an object? Can one translate well-known properties of the object into properties of the associated C*-algebra? Can invariants for the constructed C*-algebra be computed in terms of the object? Can the objects be used to classify the associated C*-algebras? At what level of generality can one use these methods to model well-known classes of C*-algebras?
My work in Algebra has been motivated by the recent development of Leavitt
path algebras, which are algebras constructed from graphs in a manner similar
to how graph C*-algebras are constructed. These Leavitt
path algebras are intimately related to graph C*-algebras, and both the
C*-algebraic and algebraic theories have guided, influenced, and assisted
each other. Indeed, each theory has had nontrivial applications to the
other, and together they have given a deeper understanding of
certain classes of algebras and C*-algebras. Much of my work in this area involves
describing the structure of Leavitt path algebras, classifying Leavitt path algebras in terms of K-theory,
examining the relationship between Leavitt path
algebras and graph C*-algebras, and using Functional Analysis results to
motivate the creation of new algebras from other objects besides graphs.