# Scottish Operator Algebras Research meeting

# University of Glasgow, 13 November, 2015

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 Mike Whittaker if you plan to come to dinner.

### Local information

Talks will be in Room 522 of the Mathematics Building. Coffee breaks will be in the maths coffee room, on the main floor of the Mathematics Building. Click here for information about getting to the Maths Building.

Our speaker Mark Lawson is unable to attend.
In his place, **Aaron Tikuisis** will give a talk on Quasidiagonality of nuclear C*-algebras.

### Speakers

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

**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)

**Ying-Fen Lin** (Queen's University Belfast) - **Positive extensions of Schur multipliers**

In my talk, I will introduce partially defined Schur multipliers and show necessary and sufficient conditions for the extension of such a multiplier to a fully defined positive Schur multiplier, in terms of an operator system canonically associated with its domain.

**Luis Santiago** (University of Aberdeen) - **\(W\otimes W\cong W\)**

The class of strongly self-absorbing C*-algebras have had an important impact on the theory of C*-algebras. Among its members we have the Cuntz algebras \(\mathcal{O}\)_{2} and \(\mathcal{O}\)_{\(\infty\)} and the Jiang-Su algebra \(\mathcal Z\). The definition of strong self-absorption uses the fact that the algebras under consideration are unital and no equivalent definition has been given to cover the nonunital case. There are however some (nonunital) stably projectionless C*-algebras that have been studied and that should belong to the corresponding class. One of them is the algebra \(W\). This algebra has trivial K-groups and hence it is considered to be a stably finite analog of the Cuntz algebra \(\mathcal O\)_{2}. This algebra first appeared in the work of Kumjian and Kishimoto, and then in the work of Jacelon.

In this talk I will show that the C*-algebra \(W\) is self-absorbing; that is, \(W\otimes W\cong W\).

**Andreas Thom** (Technische Universität Dresden) (two talks)

**Bounded Normal Generation and Invariant Automatic Continuity**

We study the question how quickly products of a fixed conjugacy class in the projective unitary group of a II_{1}-factor von Neumann algebra cover the entire group. Our result is that the number of factors that are needed is essentially as small as permitted by the 1-norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups. As an application of the techniques, we prove that every homomorphism from the projective unitary group of a II_{1}-factor to a polish SIN group is continuous. Moreover, we show that the projective unitary group of a II_{1}-factor carries a unique polish group topology. This is joint work with Philip Dowerk.

**Cantor systems and quasi-isometry of groups**

The purpose of this talk is twofold. In the first part we observe that two finitely generated non-amenable groups are quasi-isometric if and only if they admit topologically orbit equivalent Cantor minimal actions. In particular, free groups of different rank admit topologically orbit equivalent Cantor minimal actions unlike in the measurable setting. In the second part we introduce the measured orbit equivalence category of a Cantor minimal system and construct (in certain cases) a representation of this category on the category of finite-dimensional vector spaces. This gives rise to novel fundamental invariants of the orbit equivalence relation together with an ergodic invariant probability measure. This is joint work with Kostya Medynets and Roman Sauer.

**Aaron Tikuisis** (University of Aberdeen) - **Quasidiagonality of nuclear C*-algebras** (standby speaker).

I talk about quasidiagonality and joint work with S. White and W. Winter.

**Simon Wassermann** (University of Glasgow) - **Tensor primeness for certain large C*-algebras**

In 1998 Leeming Ge showed, using free entropy methods, that the group von Neumann algebra \(L\)(**\(F\)**_{2}) of the free group on 2 generators is prime, that is, it is not isomorphic to a von Neumann tensor product \(M \overline{\otimes} N\) of infinite dimensional von Neumann factors \(M\) and \(N\). More recently Ozawa has found a new proof of this result, using completely different methods and, with Popa, has obtained more general uniqueness results using these methods for tensor decompositions of the group von Neumann algebras of certain product groups.

This talk will describe recent progress on analogous questions for C*-algebras, in particular for certain group C*-algebras and the SAW*-algebras introduced by Gert Pedersen. The latter class includes all von Neumann algebras and the Calkin algebra. Ghasemi has shown that an SAW*-algebra is essentially prime in the sense that it is not isomorphic to a C*-tensor product of two infinite-dimensional C*-algebras. A short self-contained proof of this result will be given and some tantalising open problems discussed.

### Schedule

9:30-9:45 | Tea and coffee |

9:45-10:45 | Ying-Fen Lin |

10:45-11:15 | Tea and coffee |

11:15-12:00 | Aaron Tikuisis |

12:00-1:30 | Lunch |

1:30-2:15 | Andreas Thom |

2:25-3:10 | Simon Wassermann |

3:10-3:50 | Tea and coffee |

3:50-4:35 | Luis Santiago |

4:45-5:30 | Andreas Thom |

6:00 | Dinner (Oran Mor) |

### Participants

- Rauan Akylzhanov (Imperial College London)
- Getachew Alemu (Heriot-Watt University)
- Robbie Bickerton (Newcastle University)
- Yemon Choi (Lancaster University)
- David Cushing (Newcastle University)
- Daniel Estévez (Universidad Autónoma de Madrid)
- Evgenios Kakariadis (Newcastle University)
- Luke Hamblin (University of Glasgow)
- Jason Hancox (Lancaster University)
- Mark Lawson (Heriot-Watt University)
- Yin-Feng Lin (QU Belfast)
- Greg Maloney (University of Newcastle)
- David McConnell (University of Glasgow)
- Andrew Monk (University of Glasgow)
- Luis Santiago (University of Aberdeen)
- Richard Skillicorn (Lancaster University)
- Andreas Thom (TU Dresden)
- Aaron Tikuisis (University of Aberdeen)
- Richard Timoney (Trinity College Dublin)
- Gabriele Tornetta (University of Glasgow)
- Christian Voigt (University of Glasgow)
- Simon Wassermann (University of Glasgow)
- Mike Whittaker (University of Glasgow)
- Stuart White (University of Glasgow)
- Joachim Zacharias (University of Glasgow)
- Ping Zhong (Lancaster University)

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.