UK Virtual Operator Algebras Seminar: Approximating traces: what, why and how.
Christopher Schafhauser (University of Nebraska, Lincoln)
Thursday 14th May, 2020 16:45-17:30 Contact Mike.Whittaker@glasgow.ac.uk for Zoom info
Traces have played a fundamental role in operator algebras every since the beginning of the subject starting with Murray and von Neumann’s study of finite von Neumann algebras and the classification of projections in such algebras via the centre-valued trace. In Connes’s study of injective II_1-factors, he considered a certain finite-dimensional approximation property of the trace (now known as amenability) and showed a II_1-factor is injective if and only if the trace is amenable – this equivalence is one of the major steps in his proof that injective implies hyperfinite. Several related approximation conditions were introduced and studied by N. Brown in the early 2000’s. His notion of a quasidiagonal trace, which is motivated both by Connes’s notion of an amenable trace and by Halmos’s notion of a quasidiagonal operator, has proven to be especially important in the structure of simple nuclear C*-algebras and plays a key role in the classification of such algebras through the work of Elliott, Gong, Lin, and Niu. I will discuss approximation properties of traces (in particular amenability and quasidiagonality) and the relations between them.