PhD Defense - Henning Olai Milhøj

Title: Quasitraces, Tracial States, and Kaplansky’s Conjecture


This thesis examines various notions of traces on C*-algebras with a focus on quasitraces and tracial states, and on the connection between them as exemplified by Kaplansky's conjecture. Different already known characterisations of when unital C*-algebras admit quasitraces and tracial states are studied, and these characterisations are used in order to construct new numerical invariants, which informally measures the failure to admit quasitraces resp. tracial states, and which have interesting asymptotic behaviour. In particular, following a construction due to Rørdam, it is shown that for any natural number n there exists a separable, unital, nuclear, simple C*-algebra A_n such that n is the smallest integer satisfying that M_n(A_n) is properly infinite. By taking an ultraproduct of this sequence of C*-algebras without quasitraces, one obtains an example of a unital non-exact C*-algebra with a quasitrace. It is unresolved whether this quasitrace is a tracial state. We discuss the possible implications for Kaplansky's conjecture, and also how the aforementioned numerical invariants may provide more information about the conjecture. Inspired by work of Robert--Rørdam, we also provide a way of viewing the existence of C*-algebras with tracial states not approximable by limit tracial states by introducing the notion of almost tracial states. 

Moreover, this thesis presents the original as well as an alternative proof of the result due to Haagerup that quasitraces on unital, exact C*-algebras are tracial states. We emphasise the usages of the AW*-completion, which gives a link between Kaplansky's original conjecture and the modern formulation, and we also emphasise when exactness is used. The alternative proof uses the result due to Haagerup--Thorbjørnsen that C*_{r}(mathbb{F}_{infty}) is an MF-algebra, which allows for matrix approximations of certain self-adjoint elements in C*_{r}(mathbb{F}_{infty}). Using a characterisation of non-traciality of unital C*-algebras due to Haagerup will then allow us to prove that the minimal tensor product A otimes C*_{r}(mathbb{F}_{infty}) is properly infinite, and by invoking exactness and the aforementioned self-adjoint lifts, one obtains the existence of a natural number  for which M_n(A) is properly infinite. We also show that the proof strategy does not hold universally for non-exact C*-algebra by showing that, for any choice of matrix approximation of the self-adjoint elements in C*_{r}(mathbb{F}_{infty}), there exists a non-exact C*-algebra A for which the proof fails. 

In the thesis, we also initiate the study of when C*-algebras have the property that all quotients admit faithful tracial states. We provide a sufficient and necessary condition for when C*-algebras admit faithful tracial states in terms of the existence of stable ideals by using regularity properties of Cuntz semigroups. By applying this result to the quotients, we then obtain an an equivalent formulation for all quotients to admit faithful tracial states, and we use this to determine when C*-algebras are strongly quasidiagonal. 

Download the thesis.

Advisor: Professor Mikael Rørdam, Department of Mathematical Sciences, University of Copenhagen

Assessment committee:

Chair, professor Søren Eilers, Math. Copenhagen University

Professor, Marius Dadarlat, Purdue University

Assistant professor, James Gabe, University of Southern Denmark