Noncommutative Geometry and Topology in Prague

Tracially complete C*-algebras: Tracial Transfer and Structure

Tracially complete C*-algebras have been playing an increasingly central role in the latest developments on the classification of stably finite C*-algebras. In this talk I will give a quick introduction to these structures, which broadly speaking resemble bundles over closed convex sets whose fibers are von Neumann algebras. I will then proceed to show how, under suitable conditions, this intuition translates into a powerful transfer phenomenon, which allows to show that numerous properties holding for von Neumann algebras are also true for tracially complete C*-algebras. I will conclude analyzing certain structural aspects of these objects, focusing in particular on how certain key properties, such as being factorial or being a trivial W*-bundle, are not preserved when going to ultraproducts.