Run by the New Zealand Mathematics Research Institute
NZMRI 2014 Summer School:
Operator Algebra
13 – 17 January 2014         Te Anau, New Zealand
Hosted by:

## Abstracts

### Ruy Exel

Partial actions

After introducing the notion of partial actions of groups on sets, topological spaces, algebras and C*-algebras, I will discuss the notion of partial crossed product. This construction produces an algebra from a given partial action and it so happens that many of the most important C*-algebras studied in the last 35 years or so may be described as the partial crossed product relative to a partial action of a group on a topological space. Among these we mention AF-algebras, Bunce-Deddens algebras, Cuntz-Krieger algebras, Graph algebras, Wiener-Hopf algebras for quasi-lattice ordered groups (Nica), Cuntz-Krieger algebras for infinite matrices (E., Laca), the Bost-Connes algebra, the algebras A[R] associated with integral domains (Cuntz, Li), C*-algebras associated to right-angled Artin groups (Crisp, Laca) and the algebras $O_{n,m}$ associated to dynamical system or type (n,m) (Ara, E.).

Observing that partial crossed product algebras are always graded algebras one may ask whether or not a given graded algebra may be described as a partial crossed product. The main goal of this set of lectures will be to give a positive answer in case the graded algebra satisfies a certain stability property.

### Jesse Peterson

Vanderbilt University

Lecture 1: Completely positive maps and conditional expectations

One of the most useful techniques for comparing operator algebras is to look at the space of completely positive maps between them. Many analytic properties can be transferred from one algebra to the other though an appropriate completely positive map. In this lecture I will discus basic concepts in this direction and I will also discuss conditional expectation in von Neumann algebras.

Lecture 2: Injective factors

In this lecture we discuss a distinguished class of von Neumann algebras which play a central role in the theory. We will introduce the notion of an injective factor and discuss examples such as the hyperfinite II$_1$ factor, and Powers’ factors. We will also discuss some of the groundbreaking work of Connes for this class.

Lecture 3: Noncommutative Poisson boundaries

The Poisson kernel gives an isomorphism between the space of bounded harmonic functions on the unit disc, and the space of essentially bounded functions on the unit circle. Generalizing this situation, Furstenberg introduced the notion of a Poisson boundary for an arbitrary locally compact group with a distinguished probability measure. Generalizing the situation even further, Izumi defined the notion of a Noncommutative Poisson boundary of a completely positive map. We will introduce Izumi’s notion and discuss some applications in the setting of II$_1$ factors.

### Sorin Popa

University of California

Classification and rigidity in operator algebras arising from free groups

Higman has shown in 1939 that isomorphic group algebras $\Bbb C \Gamma$ of torsion free orderable groups $\Gamma$ can be isomorphic only if the groups are isomorphic. But letting $\Bbb C \Gamma$ act on the Hilbert space $\ell^2 \Gamma$ by left convolution and then taking closure in the weak operator topology, gives rise to much larger algebras, denoted $L(\Gamma)$, that tend to forget the group $\Gamma$, for instance $L(\Bbb Z \wr \Bbb Z^n)$, $n \geq 1$ are all isomorphic. The study of these operator algebras, called von Neumann algebras, was initiated in the early 1940s. A famous problem going back to that time is whether the von Neumann algebras $L(\Bbb F_n)$, associated with the free groups on $n$ generators, are non-isomorphic for different $n$’s. While this is still open, its “group measure space” version, asking whether the crossed product von Neumann algebras $L^\infty(X)\rtimes \Bbb F_n$ arising from free ergodic probability measure preserving actions $\Bbb F_n\curvearrowright X$ are non-isomoprphic for $n= 2, 3, ...$, independently of the actions, has recently been settled by Stefaan Vaes and myself. During my talks, I will comment on this result, as well as on some related problems.

### Aidan Sims

University of Wollongong

Graph C*-algebras

Over the last twenty years or so, there has been immense interest in the class of C*-algebras known as graph C*-algebras: C*-algebras that are modelled on, and encode, the connectivity structure of directed graphs. Because directed graphs themselves are fairly elementary combinatorial objects, the associated C*-algebras are quite tractable, and we know by now how to read a great deal of the internal structure of a graph C*-algebra C*(E) from the combinatorial structure of E. Nevertheless, the theory is complicated enough that there are still natural-seeming questions about graph C*-algebras that remain unresolved: for example, how do we decide, given directed graphs E and F, whether C*(E) and C*(F) are isomorphic? In my lectures I will describe how the C*-algebra of a directed graph is defined and constructed, and then give a flavour of some of the fundamental results in the subject.