# Scottish Operator Algebras Research meeting

# University of Glasgow, Friday 2 and Saturday 3 December, 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 Michael Whittaker if you plan to come to dinner, which will be held at Òran Mór.

This meeting is joint with the LMS Celebrating New Appointments meeting **"Topological dynamics and operator algebras"**, organised by Michael Whittaker.

### Local information

The meeting will take place at the University of Glasgow, in Mathematics 204. Click here for information about getting to the Maths Building.

### Speakers

Click on the talk title for the slides (where applicable).

**Jean Bellissard** (Georgia Institute of Technology and University of Münster) - **Spectrum of self-adjoint operators on aperiodic tilings: periodic approximations**

After a list of motivations for this problem, various concepts will be introduced: (i) continuous fields of self-adjoint operators, (ii) continuous field of C*-algebras, (iii) continuous fields of groupoids. Continuity of the spectrum will follow. Then using the concept of tautological groupoid, it will be shown how to control periodic approximations. In the case of one-dimensional system with finite local complexity a complete characterization of periodically approximable systems will be provided.

**Sam Evington** (University of Glasgow) - **W*-bundles**

W*-bundles were first introduced by Ozawa, motivated by work on the Toms-Winter Conjecture and, more generally, the classification of C*-algebras. I will begin with a brief introduction to W*-bundles, explaining how they combine the measure theoretic nature of tracial von Neumann algebras with the topological nature of C*-algebras. I will then discuss the relationship between the triviality problem for W*-bundles and the Toms-Winter Conjecture. Finally, I will present my work with Ulrich Pennig on locally trivial W*-bundles, culminating in the result that a locally trivial W*-bundle whose fibres all are isomorphic to the hyperfinite II\(_1\) factor \(\mathcal{R}\) is trivial.

**Elizabeth Gillaspy** (University of Münster) - **Wavelets and spectral triples for higher-rank graphs**

In joint work with Farsi, Kang, and Packer, we have constructed a representation of a higher-rank graph C*-algebra C*(\(\Lambda\)) on L\(^2\)(\(\Lambda^\infty\), M), where \(\Lambda^\infty\) is the space of infinite paths in the higher-rank graph \(\Lambda\) and M is a canonical Borel measure on \(\Lambda^\infty\). This representation gives rise to a wavelet-type decomposition of L\(^2\)(\(\Lambda^\infty\), M); in joint work with Farsi, Kang, Julien, and Packer, we have discovered that this wavelet-type decomposition is closely related to the spectral triples and Dirac operators on \(\Lambda^\infty\) studied by Pearson and Bellissard and by Julien and Savinien.

In this talk, we will explain these connections, and (time permitting) also describe the relationship between the FGKP wavelet decomposition and another spectral triple for C*(\(\Lambda\)), which was first described by Consani and Marcolli for the Cuntz algebra O\(_N\).

**Heiko Gimperlein** (Heriot-Watt University) - **Dixmier traces of nonmeasurable commutators**

As the noncommutative analogue of differential forms, commutators play a fundamental part in noncommutative geometry. In order to understand the Hölder and Lipschitz algebras of a spectral triple, as well as topological invariants of nondifferentiable maps, in this talk we discuss commutators with nonsmooth functions on Riemannian manifolds. We obtain integral formulas related to Connes' residue trace theorem and relate to classical works by Rochberg and Semmes. On the circle, a large class of non-measurable Hankel operators is obtained from Hölder continuous functions, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from circles to Riemannian manifolds, contact manifolds and noncommutative tori. Joint with Magnus Goffeng.

**Chris Heunen** (University of Edinburgh) - **Functorial spectra and discretization of C*-algebras**

Much of the structure of a noncommutative C*-algebra is determined by the collection of its commutative C*-subalgebras. However, one can rigorously prove that it does not provide a full invariant. This question is answered by the notion of an active lattice, which adds the information of how different commutative C*-subalgebras relate to each other dynamically. After a survey of this area, I will discuss how one can reconstruct an AW*-algebra from its active lattice. Time permitting I will talk about extending this from AW*-algebras to C*-algebras, leading to so-called discretizations of a C*-algebra.

**Nadia Larsen** (University of Oslo) - **KMS states on right LCM semigroup C*-algebras** (part of "Topological dynamics and operator algebras")

C*-algebras that reflect algebraic structure of semigroups through families of isometries have been studied for a long time. In the past decade, motivated by examples arising in number theory, a general construction of C*-algebras associated to left cancellative semigroups was proposed by Li, and many new examples were found. Brownlowe, Ramagge, Robertson and Whittaker coined the notion of right LCM semigroup, and supplied large classes of examples by means of the Zappa-Szep product. One remarkable feature of C*-algebras that contain families of isometries is that they tend to admit "canonical" time evolutions and interesting structure of associated KMS, or equilibrium states. This has been the case for many classes of examples. Here we investigate the KMS state structure for natural dynamics in the general setup of the full C*-algebra of a right LCM semigroup. We identify internal structural properties of the semigroup, already predicted by the boundary quotient diagram of Stammeier, governing behaviour of the KMS states. This is joint work with Afsar, Brownlowe and Stammeier.

**Xin Li** (Queen Mary University of London) - **Rigidity in C*-algebras and topological dynamics** (part of "Topological dynamics and operator algebras")

This talk is about Cartan subalgebras in C*-algebras and continuous orbit equivalence for topological dynamical systems. We explain how these notions build bridges between C*-algebras, topological dynamics, and geometric group theory. Moreover, we discuss rigidity phenomena in both settings.

**Michael Whittaker** (University of Glasgow) - **KMS states on self-similar groupoid actions** (part of "Topological dynamics and operator algebras")

A self-similar groupoid action (G,E) consists of a faithful action of a groupoid G on the path space of a graph which displays a notion of self-similarity. In this talk I will explain this concept and construct a Cuntz-Pimsner algebra from (G,E). I will then consider KMS states associated with these C*-algebras. This talk is based on joint work with Marcelo Laca, Iain Raeburn, and Jacqui Ramagge.

### Schedule

All talks are in Room 204 of the Mathematics building. Breaks and the reception will be held in the Mathematics Common Room.

**Friday 2 December**

12:30-13:20 | Chris Heunen |

13:30-14:20 | Nadia Larsen |

14:20-15:00 | Tea, coffee, and discussion |

15:00-15:50 | Xin Li |

16:00-16:50 | Michael Whittaker |

17:00-18:00 | Reception (for "Topological dynamics and operator algebras") |

18:30 | Dinner - Oran Mor |

**Saturday 3 December**

9:30-10:20 | Heiko Gimperlein |

10:20-11:00 | Tea, coffee, and discussion |

11:00-11:50 | Jean Bellissard |

11:50-13:20 | Lunch |

13:20-14:00 | Sam Evington |

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

14:30-15:20 | Elizabeth Gillaspy |

### Participants

- Rob Archbold (University of Aberdeen)
- Jean Bellissard (Georgia Institute of Technology and University of Münster)
- Alexander Belton (Lancaster University)
- Robbie Bickerton (Newcastle University)
- Joan Bosa (University of Glasgow)
- Sarah Browne (University of Sheffield)
- Yemon Choi (Lancaster University)
- Michael Dreher (Heriot-Watt University)
- Sam Evington (University of Glasgow)
- Ruaridh Gardner (University of Aberdeen)
- Dimitrios Gerontogiannis (University of Glasgow)
- Elizabeth Gillaspy (University of Münster)
- Heiko Gimperlein (Heriot-Watt University)
- Luke Hamblin (University of Glasgow)
- Chris Heunen (University of Edinburgh)
- Bence Horvath (Lancaster University)
- Evgenios Kakariadis (Newcastle University)
- Nadia Larsen (University of Oslo)
- Mark Lawson (Heriot-Watt University)
- Xin Li (QMU London)
- Ying-Fen Lin (Queen's University Belfast)
- Andrew McKee (Queen's University Belfast)
- Andrew Monk (University of Glasgow)
- Ismail Ozkaraca (University of Glasgow)
- Gábor Szabó (University of Aberdeen)
- Aaron Tikuisis (University of Aberdeen)
- Christian Voigt (University of Glasgow)
- Simon Wassermann (University of Glasgow)
- Steven Watson (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, Stuart White, and Michael Whittaker.

### Support

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