# Scottish Operator Algebras Research meeting

# University of Glasgow, Friday 24 and Saturday 25 March, 2017

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.

Funding to support Ph.D. students has already been allocated.

### Local information

The meeting will take place at the University of Glasgow, in the Boyd Or Building, the WILT Lecture Theatre, and the Maths Building (see the schedule below), and with breaks and the reception in the Mathematics Common Room. Click here for information about getting to the Maths Building. The Boyd Or Building is next to the Maths Building (to the west, also on University Avenue).

### Speakers

**Sarah Browne** (University of Sheffield) - ** A quasi-spectrum for asymptotic morphisms of graded C*-algebras**

Homotopy classes of certain asymptotic morphisms form the E-theory groups. My talk will start by equipping the set of asymptotic morphisms with the notion of a quasi-topology, as of Dardalat-Meyer. I will then define a collection of quasi-topological spaces using this and show it forms a spectrum associated to the E-theory groups. If time allows, I hope to state the product structure that comes with this construction.

**Jan Cameron** (Vassar College) - **A Galois correspondence for crossed products of C*-algebras by discrete group**

If a discrete group G acts by outer automorphisms on a unital C*-algebra A, then any subgroup H of G gives rise to a C*-subalgebra A\(\rtimes_{\alpha, r}\)H of A\(\rtimes_{\alpha, r}\)G containing A. When these are all the C*-algebras between A and A\(\rtimes_{\alpha, r}\)G, we say that a Galois correspondence holds for the inclusion A \(\subseteq\) A\(\rtimes_{\alpha, r}\)G. When A is unital and simple, such a Galois correspondence has been established for an abelian group G by Landstad, Olesen, and Pedersen; and when G is finite, by a result of Izumi. In this talk we discuss recent joint work with Roger Smith, in which we generalize these results to the case of an arbitrary discrete group G.

**Erik Guentner** (University of Hawai'i at Manoa) - **On K-amenability of CAT(0) cubical groups**

A group which act properly on a CAT(0) finite dimensional cubical complex, while not necessarily amenable, satisfies several weak forms of amenability: such a group is a-T-menable, weakly amenable and K-theoretically amenable. In the talk, based on joint work with J. Brodzki and N. Higson, I will describe a proof of K-amenability which finds its roots in earlier work of P. Julg and A. Valette on groups acting on trees.

**Simon Henry** (College de France, Paris) - **The convolution algebra of a 'locally compact' topos**

I will explain a new construction that attach an involutive algebra (or a C*-algebra) to a topos satisfying certain "local separation" and "local compactness" conditions.

The construction is in practice very close to the construction of groupoids C*-algebras and generally produce the same results up to Morita equivalence, but it shows that the associated algebra has a universal property in terms of the topos and gives a different perspective on the algebra. It also has some surprising connection with a form of Verdier duality that I will explain if I have enough time. To some extent it is a re-formulation of the construction of the C*-algebra of an étale groupoid which happen to be able to deal with some non étale groupoid.

I will not assume any knowledge of topos theory, and try as much as possible to give an intuitive understanding of what are toposes and how they are used here.

**Vaughan Jones** (Vanderbilt University) - **Do all subfators arise in quantum field theory?**

After a quick review of the state of the art in subfactor theory we will describe the Doplicher Haag Roberts construction of subfactors and attempt to address the question of whether all subfactors can be constructed somehow from DHR ones.

**Andy Monk** (University of Glasgow) - **An analogue of the Baum-Connes Conjecture for SL\(_q(2, \mathbb{C})\)**

The Baum-Connes conjecture gives a description of the K-theory of the reduced group C*-algebra of a locally compact, second countable group. In the case of a connected Lie group G, it is possible to reformulate the conjecture in terms of a deformation of G provided by a continuous field of C*-algebras. In this talk, we will look at this reformulation and report on work being done to produce an analogue of the conjecture for the quantum group SL\(_q(2, \mathbb{C})\).

**Karen Strung** (IMPAN, Warsaw) - **Group actions on Smale spaces and their C*-algebras**

A Smale space is a type of hyperbolic dynamical system (X, \(\varphi\)) which includes such well-known examples as the shifts of finite type, hyperbolic toral automorphisms and Anosov diffeomorphisms. Each Smale space gives rise to topological equivalence relations coming from the stable, unstable, and homoclinic relations. From these one may construct C*-algebras. If there is a group action on X that commutes with \(\varphi\), this induces an action on each of these C*-algebras, which in turn allows one to construct crossed product C*-algebras. I will discuss my recent work with Robin Deeley which studies properties of groups actions on X that allow us to deduce structural properties of the resulting crossed products.

**Gábor Szabó** (University of Aberdeen) - **Ocneanu-type uniqueness for certain group actions on strongly self-absorbing C*-algebras**

The classification problem for amenable group actions on injective factors emerged as a natural aim after Connes’ spectacular classification results in the 70s, which involved the classification of cyclic group actions on type II factors. After Jones’ breakthrough for finite group actions, it was Ocneanu who settled the type II case for all amenable groups in the 80s, in particular verifying that every amenable group has a unique cocycle conjugacy class for outer actions on the hyperfinite II_{1}-factor. Viewing strongly self-absorbing C*-algebras as the natural analogs of the hyperfinite II_{1}-factor, it is natural to ask whether Ocneanu’s result can hold its ground in this setting. That is, for a given amenable group G and a strongly self-absorbing C*-algebra D, is there a unique cocycle conjugacy class for strongly outer G-actions on D? It is by now well-known that such rigid behavior is obstructed by torsion in G, but I will argue why the answer to the aforementioned question could be ‘Yes’ for torsion-free groups. I will explain how the theory of strongly self-absorbing actions can be employed to settle this problem in the abelian case. In particular, the resulting uniqueness theorem is new even for Z^{3}-actions on the Jiang-Su algebra.

### Schedule

Breaks and the reception will be held in the Mathematics Common Room.

**Friday 24 March**

Talks are in Room 513 of the Boyd Or Building, except for the EMS lecture which is in the Western Infirmary Lecture Theatre (WILT).

13:00-13:50 | Gábor Szabó |

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

14:30-15:20 | Simon Henry |

15:30-16:00 | Andy Monk |

16:00-16:30 | Tea, coffee, and discussion |

16:30-17:30 | EMS lecture by Vaughan Jones |

18:30 | Dinner - Òran Mór |

**Saturday 25 March**

Talks are in Room 515 of the Mathematics Building.

9:30-10:20 | Karen Strung |

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

11:00-11:50 | Vaughan Jones |

12:00-12:30 | Sarah Browne |

12:30-14:00 | Lunch |

14:00-14:50 | Erik Guentner |

15:00-15:50 | Jan Cameron |

### Participants

- Rob Archbold (University of Aberdeen)
- Sarah Browne (University of Sheffield)
- Jan Cameron (Vassar College)
- Sam Evington (University of Glasgow)
- Ruaridh Gardner (University of Aberdeen)
- Dimitrios Gerontogiannis (University of Glasgow)
- Heiko Gimperlein (Heriot-Watt University)
- Erik Guentner (University of Hawai'i at Manoa)
- Luke Hamblin (University of Glasgow)
- Jason Hancox (Lancaster University)
- Simon Henry (College de France, Paris)
- Chris Heunen (University of Edinburgh)
- Bence Horvath (Lancaster University)
- Vaughan Jones (Vanderbilt University)
- Evgenios Kakariadis (Newcastle University)
- Mark Lawson (Heriot-Watt University)
- Andrew Monk (University of Glasgow)
- Ismail Ozkaraca (University of Glasgow)
- Karen Strung (IMPAN, Warsaw)
- Gábor Szabó (University of Aberdeen)
- Aaron Tikuisis (University of Aberdeen)
- Christian Voigt (University of Glasgow)
- Simon Wassermann (University of Glasgow)
- Jared White (Lancaster University)
- 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.