# Scottish Operator Algebras Research meeting

# University of Aberdeen, Wednesday 16 and Thursday 17 March, 2016

This operator algebras meeting is one in a series of meetings in Scotland focusing on operator algebras research. Participants are welcome from anywhere. Please email Aaron Tikuisis if you plan to come to dinner.

Participants may be interested in the Edinburgh Mathematical Society meeting, held at the University of Dundee on Friday 18 March, 2016. Click here for details.

### Local information

Talks will be in Fraser Noble 156 (the Maths Seminar Room) and 185 (see the Schedule below). Coffee breaks will be in the maths lounge, just outside the seminar room. Click here for information about getting to and around campus.

### Speakers

**Jorge Castillejos Lopez** (University of Glasgow) - **Colouring C*-algebras**

In this talk I will introduce a coloured equivalence between C*-algebras. This notion is motivated by the fact that regularity properties of simple nuclear C*-algebras correspond to "coloured" properties of injective von Neumann algebras.

**Robin Hillier** (Lancaster University) - **Loop groups and noncommutative geometry**

Loop groups are infinite-dimensional Lie groups, with connections to various areas of mathematics and physics, and with an interesting representation theory. We describe the so-called positive-energy representation theory in terms of operator algebraic K-theory and noncommutative geometry. The is done constructively, using ideas from conformal field theory.

**Mark Lawson** (Heriot-Watt University) - **Non-commutative Stone dualities**

In Renault's famous monograph on constructing C*-algebras from
étale topological groupoids another class of algebraic structures also intrudes for somewhat mysterious reasons:
inverse semigroups. Not long afterwards, Kumjian also used inverse semigroups to construct C*-algebras.
He regarded them as being non-commutative bases (of presumably non-commutative topological spaces, whatever they might be).
If you read papers by Ruy Exel inverse semigroups litter the place.
What is going on? Why should inverse semigroups be of any interest to serious operator algebra theorists?
I am not an operator algebraist but, as the saying goes, I know people who are.
In this talk, I shall explain from scratch the connection between inverse semigroups
and étale topological groupoids and include some examples to illustrate the striking parallels with C*-algebras
— there are for instance, AF inverse monoids and Cuntz inverse monoids.
This is work that has been carried out in collaboration with
Johannes Kellendonk (Lyon), Ganna Kudryavtseva (Ljubljana), Daniel Lenz (Jena),
Stuart Margolis (Bar Ilan), Pedro Resende (Lisbon), Phil Scott (Ottawa) and Ben Steinberg (CUNY)

**Martin Mathieu** (QU Belfast) - **Towards a sheaf cohomology theory for C*-algebras**

This is a preliminary report on work in progress with Pere Ara (Barcelona).
On the basis of our sheaf theory for C*-algebras, we intend to develop a full
sheaf cohomology theory with the ultimate aim to define cohomological dimension
and possibly new invariants for C*-algebras. As the categories (of operator modules)
to be used are not abelian, the usual techniques don't apply and various steps (will)
have to be done via other methods.

**Christian Skau** (NTNU Trondheim) - **Ordered Bratteli diagrams and Cantor minimal systems**

Simple dimension groups (G,G^{+},u), with distinguished order unit u, can be defined in three equivalent ways, and we list them in the order they were historically introduced:

(i) Via Bratteli diagrams.

(ii) Abstractly, as (unperforated) ordered abelian groups satisfying the
Riesz interpolation property.

(iii) Dynamically, via Cantor minimal systems.

It is well known that simple dimension groups appear as complete isomorphism invariants for (simple) AF-algebras as well as for C*-crossed
products associated to Cantor minimal systems. Furthermore, simple dimension groups also appear as complete invariants for orbit equivalence,
respectively, strong orbit equivalence, of Cantor minimal systems. In this talk we will mention some fairly recent results how change of the ordering of a given Bratteli diagram yield entirely different Cantor minimal systems, while the systems themselves are orbit equivalent, respectively, strong orbit
equivalent. We will also give examples of special — yet very comprehensive — classes of Cantor minimal systems that are assoiated to "nice" ordered
Bratteli diagrams.

**Jack Spielberg** (Arizona State University) - **C*-algebras associated to graphs of groups**

There are many interesting examples of groups acting on trees, arising in
various fields (e.g. combinatorial group theory, number theory, geometry). When a
group acts on a tree, it necessarily also acts on the boundary of the tree, a (totally
disconnected) compact Hausdorff space. The C*-algebras obtained from the crossed
product construction include many fundamental examples. I will describe methods for
analyzing such crossed products, developed in joint work with Nathan Brownlowe,
Alex Mundey, David Pask and Anne Thomas.

**Christian Voigt** (University of Glasgow) - **Plancherel theorem for complex quantum groups** (joint with R. Yuncken)

The aim of this talk is to discuss an explicit formula for the Plancherel measure of standard deformations of complex semisimple Lie groups. I'll start by explaining some background on the classical Plancherel formula and its generalizations, including work of Duflo-Moore and Harish-Chandra's Plancherel formula for complex groups.

### Schedule

**Wednesday 16 March**

Talks in FN156.

13:00-14:00 | Mark Lawson |

14:00-14:30 | Tea, coffee, and discussion |

14:30-15:15 | Christian Skau |

15:30-16:15 | Christian Voigt |

16:15-17:00 | Tea, coffee, and discussion |

17:00-17:45 | Jack Spielberg |

19:00 | Dinner - Christos Greek Taverna |

**Thursday 17 March**

Talks in FN185.

9:00-9:45 | Jorge Castillejos Lopez |

9:45-9:55 | Tea, coffee, and discussion |

9:55-10:55 | Robin Hillier |

11:00-13:00 | Tea, coffee, discussion, and lunch |

13:00-14:00 | Martin Mathieu |

14:15-15:00 | Christian Skau |

15:00-15:30 | Tea, coffee, and discussion |

15:30-16:15 | Jack Spielberg |

### Participants

- Rob Archbold (University of Aberdeen)
- Alexander Belton (Lancaster University)
- Joan Bosa (University of Glasgow)
- Sarah Browne (University of Sheffield)
- Jorge Castillejos Lopez (University of Glasgow)
- Sam Evington (University of Glasgow)
- Heiko Gimperlein (Heriot-Watt University)
- Luke Hamblin (University of Glasgow)
- Chris Heunen (University of Edinburgh)
- Robin Hillier (Lancaster University)
- Mark Lawson (Heriot-Watt University)
- Martin Mathieu (Queen's University Belfast)
- David McConnell (University of Glasgow)
- Andrew Monk (University of Glasgow)
- Christian Skau (NTNU Trondheim)
- Jack Spielberg (Arizona State University)
- Aaron Tikuisis (University of Aberdeen)
- Gabriele Tornetta (University of Glasgow)
- Christian Voigt (University of Glasgow)
- Simon Wassermann (University of Glasgow)
- Stuart White (University of Glasgow)
- Mike Whittaker (University of Glasgow)
- Joachim Zacharias (University of Glasgow)

The meeting is organised by Aaron Tikuisis and Stuart White.

## Support

Funding for this meeting is provided by the Glasgow Mathematical Journal Trust and the Edinburgh Mathematical Society.