Research
C*-ALGEBRAS
A C*-algebra is a mathematical structure generalizing that of the bounded linear operators between Hilbert spaces, B(H). As such, it has far reaching consequences for quantum mechanics. Due to remarkable work by Gelfand and Naimark we know that every commutative C*-algebra arises as an algebra of continuous functions on a locally compact space. Hence, the study of C*-algebras is in some sense the study of non-commutative topology. In general, every C*-algebra can be imbedded inside B(H) for some Hilbert space H.
CARTAN SUBALGEBRAS
A Cartan subalgebra of a C*-algebra is a distinguished type of maximally Abelian subalgebra. Examples include diagonal subalgebras of matrix algebras, the function space subalgebra of certain crossed product C*-algebras, and many more.
There is an interplay between C*-algebras, dynamics, and group theory via Cartan subalgebras. Additionally, Cartan subalgebras have recently appeared in the classification programme for C*-algebras. This programme, due to many hands, aims to classify C*-algebras via their K-theoretic and tracial data. It turns out that every classifiable C*-algebra has a Cartan subalgebra.
The question of existence of Cartan subalgebras has also proven vital in this regard. An important open problem is whether every nuclear separable C*-algebra satisfies the UCT (this is known as the UCT problem). This problem has had reductions to an existence question about Cartan subalgebras.
TOPOLOGICAL GROUPOIDS
Groupoids are algebraic structures that generalize the notion of a group; indeed multiplication in a groupoid does not need to be defined on all pairs of elements. Algebraically, groupoids are nothing more than a union of products of equivalence relations with groups. Topologically however, groupoids are more interesting. From a topological groupoid with certain properties we can construct C*-algebras, and many C*-algebras arise from topological groupoids (in this case we say the C*-algebra has a groupoid model). Every C*-algebra with a Cartan subalgebra has a (twisted) groupoid model. Hence all classificable C*-algebras have groupoid models.
For a concise introduction to these topics, see the Preliminaries of my PhD thesis.
