In my talk, I will discuss a new result providing a positive answer to a natural problem about continuous families of projections in II_1-factors. The problem is naturally viewed through the lens of “W*-bundles”, and our proof is via a novel technique which utilises free probability theory in a uniform manner across these bundles. This leads to the notion of selflessness for W*-bundles, which also provides a number of other regularity properties for these objects, such as strict comparison, real rank zero, and stable rank one. This is joint work with David Jekel and Stuart White.

In 2008, Toms constructed a counterexample to the Elliott conjecture: a pair of simple, separable, nuclear and unital C*-algebras which are indistinguishable by the Elliott invariant, but are not isomorphic. The key to distinguishing this pair of carefully crafted C*-algebras lies with a rather refined invariant called the Cuntz semigroup. Consequently, Toms’s counterexample highlighted the importance of the Cuntz semigroup to the classification of C*-algebras.

In this talk, we will discuss the Cuntz semigroup in the context of graph C*-algebras, a highly diverse class of mostly non-simple C*-algebras. In particular, we will accentuate how the highly organised structure of a unital graph C*-algebra is reflected in its Cuntz semigroup and if enough time permits, mention properties of unital graph C*-algebras that are revealed by these Cuntz semigroups.

Much recent research has gone into understanding the first order theory of II1 factors. Very recently, Peterson released a preprint which develops deformation rigidity in the ultrapower setting. His techniques give many explicit examples of non-isomorphic ultrapowers for natural families of II1 factors. In this talk, I will introduce some of Peterson's techniques and results, including an analogue of amenability in the ultrapower setting and the interplay between property T and malleable deformations.

One of the problems posed by Kadison in 1967 asks whether every separably acting von Neumann algebra is generated by a single element. The problem remains open in its full generality but significant progress has been made since. One can of course ask the same question in the C*-algebraic setting where, however, counterexamples are abundant even among commutative C*-algebras. I will give an overview of the history of the problem and then discuss some recent results on single generation of C*-algebras associated to graphs and C*-algebras with Cartan subalgebras.

We consider the question of whether almost-homomorphisms are close to honest homomorphisms. I’ll survey a few historical results, with different source/target collections of algebras, and also consider what to take as the definition of “almost-homomorphisms”. If we end up having time, I will sketch an elementary proof that almost-characters from commutative C*-algebras are close to honest characters.