# QOP Meeting in Leeds, 2013

The third meeting of the QOP Network for the year 2012/13 will be held at the University of Leeds, 23--24 September 2013. The formal title of the meeting is "Quantum (semi)groups and (co)actions" although certainly Quantum Probability theory will feature, and there are likely to be talks from all areas of interest to the QOP Members. There is no formal registration, but interested people are welcome to email Matthew Daws.

### Programme

Available also in PDF Format.

#### Monday 23rd September

- 11-12 Frank Oertel: "Approximation properties of Banach spaces, the principle of local reflexivity for operator ideals, and factorisation of finite rank operators".
- Lunch
- 1:20-2:00 Mateusz Jurczyński: "Lie-Trotter Product Formulae in Quantum Probability"
- 2:05-3:05 Michal Gnacik: "Quantum random walks and quasifree representations of the CCR algebra"
- Tea
- 3:35-4:35 Stephen Wills: "An algebraic construction of quantum flows with unbounded generators"
- 4:40-5:40 Alexander Belton: "Feynman-Kac perturbation of quantum stochastic flows"
- Dinner is a local restuarant will be arranged as a group.

#### Tuesday 24th September

- 9:05-10:05 Steve Power: "Topics in the analysis of infinite bar-joint frameworks: from Combinatorics to Operator Algebras"
- Coffee
- 10:30-11:30 Stuart White: "Regularity properties in the classification of simple nuclear C*-algebras"
- 11:35-12:35 Biswarup Das: "Quantum Levy processes and equivariant stochastic flows"
- Lunch
- 1:50-2:50 Adam Skalski: "Haagerup property for discrete quantum groups and some connections to convolution semigroups of states"
- 3-4 Anna Wysoczanska-Kula: "Compact quantum groups related to functions on permutations"

### Location

All talks will be held in Roger Stevens LT 01. Access from Mathematics is to cross the bridge on level 10 (exit by the cafe bar) and then take the lift or stairs down to the bottom of the lecture-theatre block. The room has a blackboard and data-projector, and in a weak sense these can be used at the same time.

(Room booked 10-5 and 9-4).

Some General travel information to the School of Maths is avaliable. In particular, there is a Campus Map. On the PDF Format Map you'll find the School of Maths in Building 84 (at the east end, near 74) while the Roger Stevens building is 89.

### Lunch and dinner

There are a number of lunch options on campus. On both days, I will probably lead a group over to the refectory (building 32 on the PDF map). This serves a wide variety of hot and cold food of reasonable quality at a reasonably price. A variety of generally excellent coffee shops (selling also sandwiches, soup etc.) can be found on Woodhouse Lane directly across from the Parkinson Building (60 on the map).

On Monday night we will arrange, slightly on the spur of the moment, somewhere in the city centre to have dinner (the hope being that early on a Monday evening, accommodating circa 15 people will not be too hard).

### Accommodation options

Some cheap but good accommodation options, all in the region of £40 a night if you book early:

- Storm Jameson Court - University accommodation on campus; somewhat better than your average halls of residence! (I'm now informed this is fully booked.)
- Premier Inn, Leeds Arena - Next to the new Leeds Arena to the north end of the city centre, and a 10 minute walk to campus.
- Ibis Hotel, Leeds Centre - To the west end of the city centre, and a 15 minute walk to the campus.

For other options, you can do worse than to consult Late Rooms and see what a search finds.

## Abstracts

#### Frank Oertel

"Approximation properties of Banach spaces, the principle of local reflexivity for operator ideals, and factorisation of finite rank operators"

After a short introduction to the language of Pietsch's famous theory of operator ideals (including some illustrative examples), we will concentrate on a deepening of the relationship between (maximal) adjoint Banach ideals and conjugate Banach ideals. This leads to somehow surprising links between the existence of a suitable operator ideal norm on operator ideal products (which in general do not inherit an ideal norm from their normed factors), Grothendieck-approximation properties of Banach spaces, a transfer of the famous Principle of Local Reflexivity of Lindenstrauss, Rosenthal, Johnson and Zippin to operator ideals, and some "nice" factorisation properties of finite rank operators, formulated in the language of so called "accessible operator ideals". For example, we will recognise that also the maximal Banach ideal of absolutely 1-summing operators (and its dual) does not satisfy Floret's "m.a.p. factorization property".

Our aim is to give a rather algebraic approach by - occasionally - linking the language of general operator ideals with the no less important and powerful language of Grothendieck's tensor norms on tensor products of Banach spaces and trace duality, leading to further interesting results between certain operator ideal components, their dual spaces and approximation of finite rank operators in those operator ideal components.

Finally, we will present a few open problems.

#### Mateusz Jurczyński

"Lie-Trotter Product Formulae in Quantum Probability"

The Lie product formula expresses the exponential of A+B as the limit of the nth power of the product of the exponential of A/n with the exponential of B/n, as n tends to infinity. Here A and B are square matrices. The Trotter product formula is a far-reaching extension to the setting of one-parameter semigroups on a Banach space. Quantum stochastic cocycles provide a natural quantum probabilistic extension of one-parameter semigroups. In this talk I shall describe the known extensions of the Lie-Trotter product formula to the quantum stochastic setting. The talk includes an account of the relevant background quantum stochastic theory and ends with some glimpses of a new approach through multiple quantum Wiener integrals. (Based on joint work with Martin Lindsay.)

#### Michal Gnacik

"Quantum random walks and quasifree representations of the CCR algebra"

We present a random-walk approximation to quantum stochastic unitary operator cocycles. By employing a faithful normal state on the ``particle algebra'' -- rather than a vector state -- we obtain limiting cocycles which are strictly quasifree, thus the driving noises form a type III representation of the relevant CCR algebra. Motivation for this work arises from discrete models for the Hamiltonian description of repeated quantum interactions with a sequence of particles in a given faithful normal state. (Based on joint work with Alexander Belton and Martin Lindsay.)

#### Stephen Wills

"An algebraic construction of quantum flows with unbounded generators"

We discuss a new result on the existence of solutions to quantum stochastic differential equations. This uses necessary algebraic conditions to lessen the work required to apply our criteria compared to previous existence results. Some examples of interest will be discussed. Joint work with Alexander Belton.

#### Alexander Belton

"Feynman-Kac perturbation of quantum stochastic flows"

Quantum stochastic flows, so Markov semigroups, may be perturbed by constructing multiplier cocycles as solutions of quantum stochastic differential equations. We will explain these ideas in some detail, initially in the setting of von Neumann algebras. The problems which arise in the more general C*-algebraic situation will then be discussed and partially resolved. (Joint work with Martin Lindsay and Adam Skalski, and with Stephen Wills.)

#### Stuart White

"Regularity properties in the classification of simple nuclear C*-algebras"

Over the last 10 years, it has become apparent that not all simple separable nuclear C*-algebras are amenable to classification via K-theoretic invariants --- additional regularity properties are required for classification. In this survey talk I'll describe (some of) these regularity properties and recent work on relating them.

#### Biswarup Das

"Quantum Levy processes and equivariant stochastic flows"

Hopf *-algebras canonically associated with compact quantum groups are called CQG algebras. These are distinguished by having a translation-invariant state, called the Haar state. It is known that every quantum Levy process (i.e. pointwise-continuous, time-homogeneous, independent-increment process) on a counital *-bialgebra may be realised as a quantum stochastic process on a symmetric Fock space.

In this talk it will be shown how all quantum Levy processes on a CQG algebra are induced by unitary processes on the GNS space with respect to the Haar state. This will be applied to the study of equivariant stochastic flows on noncommutative manifolds, via quantum isometry groups. Joint work with Martin Lindsay.

#### Adam Skalski

"Haagerup property for discrete quantum groups and some connections to convolution semigroups of states"

I will describe equivalent definitions of the Haagerup property for discrete groups and its recent generalizations to discrete quantum groups. A special focus will be put on the role of conditionally negative functions, which in the quantum setting are replaced by generators of convolution semigroups of states on the dual groups. The talk is based on joint work with Matt Daws, Pierre Fima and Stuart White.

#### Anna Wysoczanska-Kula

"Compact quantum groups related to functions on permutations"

In 1989 S. L. Woronowicz suggested a method which allowed to construct the twisted SU(n) compact quantum groups (CQG). The method relies in general on a choice of a non-degenerate function on n-tuples. However, in many cases the related CQG becomes trivial (i.e. the C*-algebra is commutative). During the talk, we shall remind the method and shortly review the known examples of non-trivial CQG's comming from this construction. Then we shall try to classify the CQG's related to functions on permutations for n=3.

## Participants and their affiliations

- Alexander Belton, Lancaster University
- H. Garth Dales, Lancaster University
- Biswarup Das, University of Leeds
- Matthew Daws, University of Leeds
- Michał Gnacik, Lancaster University
- Mufida Hmaida, University of Leeds
- Mateusz Jurczyński, Lancaster University
- Zoltán Kádár, University of Leeds
- Niels Laustsen, Lancaster University
- J. Martin Lindsay, Lancaster University
- Frank Oertel, University of Southampton
- Jonathan Partington, University of Leeds
- Stephen Power, Lancaster University
- Charles Read, University of Leeds
- Adam Skalski, IMPAN, Warsaw
- Richard Skillicorn, Lancaster University
- Joshua Tattersall, University of Leeds
- Steven Trotter, University of Leeds
- Stuart White, University of Glasgow
- Stephen Wills, University College Cork
- Anna Wysoczańska-Kula, University of Wrocław