Charles Starling

In this talk I will define analogues of supports of continuous functions to general Hausdorff spaces and disjointness relations for such functions, and prove that this data completely determines locally compact spaces satisfying certain regularity conditions. This allows us to recover theorems by Kaplansky, Milgram and Banach-Stone, among others, which recover a topological space X from its (scalar-valued) continuous functions, as well as obtain new consequences related to the classification of (commutative) C*-algebras.

In this talk I'll introduce Bratteli Diagrams and present some applications of this tool in the study of AF-Algebras and Dimension Groups. Afterwards I'll show some properties of a specific Bratteli Diagram called Eulerian Graph, namely its traces and some dynamical and topological features.

Viewing random walks on groups as (infinite) Markov chains, we can ask what the infinity-valued harmonic functions are, as well as their domains. This can be reformulated in a much less awkward manner via standard dimension group constructions. Then we examine the relationship between the structure of these things and properties of the group. As a simple example, is every such a limit (in a natural sense) of global harmonic functions? The answer is no for all nonabelian torsion-free nilpotent groups and all choices of generating sets, but can be yes for some choices of sets of generators of free groups.

This talk will feature recent results on the injectivity of VN(G) as an operator module over the Fourier algebra A(G) for general locally compact quantum groups G. Contrary to the operator space category, we show that amenability of G is equivalent to injectivity of VN(G), and present a variety of applications. We will also discuss a recently introduced notion of inner amenability for quantum groups and its relation to approximation properties.

In this talk we learn about a representation theorem for bounded Borel measures on a locally compact groups, the motivation behind it, and if time permits a generalization of it. To do so, we briefly introduce Fourier algebras on locally compact groups, the operator space structure on them, and their completely bounded multipliers.

I will discuss a certain semigroup one can associate to a k-graph. Roughly speaking it makes a vertex equivalent to all vertices one edge away. This semigroup does not "see all", but it is possible that both stable finiteness and pure infiniteness of a simple k-graph C*-algebra can be characterised via stable finiteness and pure infiniteness of this semigroup. This is join work with Pask and Sims.

Herz–Schur multipliers of a discrete group have proved useful in the study of C*-algebras, as they can be used to link properties of a group to approximation properties of the associated reduced group C*-algebras. The development of these ideas has shed light on some C*-algebras properties, and motivated the introduction and study of others.

I will introduce Herz–Schur multipliers, and discuss some of their applications, before describing a generalisation of these functions to multipliers of a C*-dynamical system.

In the final part of the talk I will show how the generalised Herz–Schur multipliers can be used to study approximation properties of the reduced crossed product formed by a group acting on a C*-algebra, paralleling the applications of Herz–Schur multipliers; these new tools allow us to study approximation properties of the reduced crossed product without requiring the group to be amenable.

For a partial action of a group in a Hausdorff, locally compact, totally disconnected topological space, we realize the associated partial skew group ring as a Steinberg algebra (over the transformation groupoid attached to the partial action).

Given a partial action π of an inverse semigroup S on a ring A one may construct its associated skew inverse semigroup ring A ⋊_π S. When A is commutative, we show that the ring A ⋊_π S is simple if, and only if, A is a maximal commutative subring of A ⋊_π S and A is S-simple.

More generally than the Steinberg algebra of the transformation group, we prove that any Steinberg algebra, associated to a Hausdorff ample groupoid, can be seen as a partial skew inverse semigroup ring. As an application of our result above we present a new proof of the simplicity criterion for a Steinberg algebra.

It is well known that every C*-algebra has an increasing approximate unit w.r.t. the usual partial order on the positive unit ball. We consider the strict order << instead, where a << b means a = ab. Here again it is well known that every separable or sigma-unital C*-algebra has a <<-increasing approximate unit, but the general case remained unresolved. In this talk we outline our recent work showing that this extends to omega_1-unital C*-algebras but not, in general, to omega_2-unital C*-algebras. In particular, we consider C*-algebras defined from Kurepa/Canadian trees which are scattered and hence LF but not AF in the sense of Farah and Katsura. It follows that whether all separably representable LF-algebras are AF is independent of ZFC.

The Choquet order on measures is used to establish that states on a function system always have a representing measure supported on the set of extreme points of the state space (in a technical sense). We introduce a new operator-theoretic order on measures, and prove that it is equivalent to the Choquet order. This leads to some improvements in the classical theory, but more importantly it leads to some new operator-theoretic consequences. In particular, we establish Arveson's hyperrigidity conjecture for function systems. This yields a significant strengthening of the classical approximation theorems of Korovkin and Šaškin. This is joint work with Matthew Kennedy.

Nuclear dimension is one of several ways to get a non-commutative version of topological covering dimension and has several uses in studying C*-algebras, for instance in classification. This talk will be concerned with looking at the details of nuclear dimension, how it is defined and what properties it has, while giving details of its purpose and background.

We will discuss cross product algebras arising from dynamical systems and show that these contain nested subhomogeneous algebras in a natural way. The limit is then a large subalgebra in the sense of Phillips. The subhomogeneous algebras and their sequential embeddings feature a certain diagonal structure that we well make explicit. We can then show that any simple limit of these diagonal algebras by diagonal maps has stable rank one, which is to say that invertible elements are dense. It then follows from the theory of large subalgebras that the original cross product algebra also has stable rank one.

Roe algebras are C*-algebras naturally associated to a discrete group (or more generally a coarse space), which completely recapture the metric structure of the group. They come equipped with a natural Cartan subalgebra, and in this talk, based on joint work with Rufus Willett, I'll discuss how unique this subalgebra is. All necessary terminology will be introduced as we go.

I will show a very general formulation of a Beurling-Fourier algebra, A(G,W), for a locally compact group G. This generalizes the definition of Beurling algebras from abelian groups. Like a Fourier algebra, this is always a commutative Banach algebra. I show, how to construct these algebras, in general. Through some essential examples, I show how we can gain their spectral theory, using theory of the complexification of G.

In a wide range of mathematical problems the existence of a solution is equivalent to the existence of a fixed point for a suitable map or a critical point for an appropriate variational or hemi-variational problem. In particular, we are interested in finding such solutions which possesses certain properties. The existence theory is therefore of paramount importance in several areas of mathematics and other sciences. In this talk we shall provide a variational principle that allows us to solve problems of the general form 0 ? F(u), for a possibly multi-valued map F on a given convex set K. This variational principle has many applications in partial differential equations while unifies and generalizes several results in nonlinear Analysis such as a certain class of fixed point theorems, critical point theory on convex sets and the principle of symmetric criticality.

I will give an overview of the notion of the injective envelope and C*-envelope of an operator system or (generally nonself-adjoint) operator algebra and sketch some of the arguments.

Motivated by the goal of classifying essentially normal operators using Fredholm indices, Brown--Douglas--Fillmore (BDF) classified all extensions of the form

0 → K → E → C(X) → 0

where X is a compact subset of the plane.

Perhaps one reason for the success of the BDF theory was that in the BDF context, the multiplier algebra B(ℓ₂) and the corona algebra B(ℓ₂)/K have particularly nice structures. For example, the uniqueness of the neutral element is essentially given by the Weyl--von Neumann--Berg Theorem which, to a younger generation of specialists, implies real rank zero.

The first talk discusses the Elliott--Kucerovsky theory of absorbing extensions, an important result which solved a question raised by multiple authors and contains all the previous theories of BDF, Voiculescu, Kasparov and Lin.

The second talk is a technical talk where we discuss a Double Commutant Theorem which solves, in the stable case, a question raised by Pedersen. This will be a more technical talk.

The third talk summarizes my work on nonstable extension theory.

The three talks will be independent of each other.

Nielsen characterized the convertibility of two finite-dimensional bipartite pure states via local operations and classical communication (LOCC) using majorization. This important result, which has seen many applications in quantum information, describes the LOCC-transfer of entanglement between bipartite pure states. In this talk, we present a version of Nielsen's theorem in the commuting operator framework using a generalized class of LOCC operations and the theory of majorization in von Neumann algebras. This allows us, in particular, to witness a fundamental property of entanglement in the absence of a Hilbert space tensor product.