Structure and Classification of C* Algebras

The project "Structure and Classification of C*-algebras" falls within the EPSRC
Mathematical Analysis research area. This project aims to obtain general results and this will
be pursued both through examining key test cases and utilising recent developments in tiling
theory for amenable groups.
This is heavily motivated by the work done in the '70s by Alain Connes who proved that the
hyperfiniteness condition for von Neumann algebras is equivalent to requiring injectivity.
Precisely he obtained a strong internal approximation property. Thus, he was able to give an
almost complete classification of injective factors. Since then, using ideas from his work, the
C* algebras community tried to obtain a classification of C* algebras.
By making clever analogies, it is now known that separably acting injective factors have an
analogue class in the C*-setting, namely simple separable unital nuclear C* algebras. Through the work of many researchers, by additionally imposing two conditions, the UCT and Zstability, we have a complete classification of these algebras using K-theoretic data. However, this is still unsatisfactory since the two extra conditions are not totally understood.
One aim of this project would be to analyse the Z-stability condition for simple separable
nuclear unital C* algebras. In particular and of primary relevance to this project, we would
focus on understanding how these conditions behave under fundamental operations and one
such example is taking crossed products with discrete groups. There's a rich connection
between C* algebras and groups, with many deep conditions, such as amenability, playing an
important role. One can take different examples of groups and see how they act on a given
nuclear C* algebra, thus hopefully obtaining abstract conditions on the group and on the
action which would ensure that the Z-stability condition holds true. An equally interesting
topic is determining for what kind of groups this fails and what are the possible implications.
Right now state of the art results in this direction has quite restrictive conditions on both
traces and how the group acts on traces.
Given the change produced by the crossed product, we want to retain as much information as possible from the underlying algebra, so the study of traces shall prove indispensable. Since every trace on a C* algebra induces a finite von Neumann algebra through the GNS
construction and a II1 factor is injective if and only if every trace is amenable, by adapting
Connes' techniques, me and my supervisor, Prof. Stuart White, are hoping to understand the
structure of the newly formed algebra by exploring the effects on amenable traces. It is
tautological that any quasidiagonal trace is amenable, but the converse is not known. By
exploring more on this implication, it is possible to expand on the theory of quasidiagonal C*
algebras. A key test case is the hyperfinite II1 factor, which has a unique amenable trace. It is still an open question whether this trace is also quasidiagonal or not