Christopher Schafhauser, Department of Pure Mathematics, University of Waterloo
"An Embedding Theorem for C*-algebras"
A C*-algebra consists of an algebra of bounded linear operators acting on a Hilbert space which is closed the adjoint operation (roughly, the transpose) and is complete in a certain metric. Typical examples include the ring of n × n complex matrices and the ring C(X) of representation of continuous functions from a compact space X to the complex numbers. Many more interesting examples arise from various dynamical objects (e.g. group and group actions) and from various geometric/topological constructions.
The structure of finite dimensional C*-algebras is well understood: they are finite direct sums of complex matrix algebras. The class of approximately finite-dimensional (AF) C*-algebras, ones which may be written as (the closure of) an increasing union of finite-dimensional subalgebras, are also well understood: they are determined up to isomorphism by their module structure. However, the class of sub-algebras of AF-algebras is still rather mysterious; it includes, for instance, all commutative C*-algebras and all C*-algebras generated by amenable groups. It is a long-standing problem to find an abstract characterization of subalgebras of AF-algebras.
I will discuss this class of algebras and a recent partial answer to this AF-Embedding problem. My goal is to keep the talk accessible and mostly expository. In particular, I won't assume any prior knowledge of operator algebras or functional analysis.