# QOP Meeting in Leeds, 2015

The fifth meeting of the QOP Network will be held at the University of Leeds, 8-9 January 2015. We have no formal title, the meeting rather representing a Potpourri of research interests of the members. There is no formal registration, but interested people are welcome to email Matthew Daws.

### Programme

Confirmed speakers are:

• Gabor Elek (Lancaster)
• Marcel de Jeu (Leiden)
• Derek Kitson (Lancaster)
• Paul Mitchener (Sheffield)
• Przemysław Ohrysko (Warsaw)
• Alexey Popov (Newcastle)
• Jan Rozendaal (Delft)
• Richard Skillicorn (Lancaster)
• Simeng Wang (Warsaw and Besancon)
• Mateusz Wasilewski (Warsaw)
• Ping Zhong (Indiana and Lancaster)

The (formal) programme will begin around 11am on Thursday, and end by 4:30pm on Friday, to allow flexibility with travel.

Thursday 8th January Friday 9th January
11:10 -- 12:10Derek Kitson 9:00 -- 10:00Alexey Popov
12:15 -- 13:00Jan Rozendaal 10:05 -- 11:05Vladimir Kisil
13:00 -- 14:00Lunch 11:05 -- 11:30Coffee
14:00 -- 15:00Marcel de Jeu 11:30 -- 12:30Przemysław Ohrysko
15:05 -- 16:05Gabor Elek 12:30 -- 1:30Lunch
16:05 -- 16:35Tea 1:30 -- 2:15Richard Skillicorn
16:35 -- 17:35Paul Mitchener 2:20 -- 3:05Simeng Wang
17:35 -- 18:20Mateusz Wasilewski 3:05 -- 3:30Tea
18:20To pub and dinner 3:30 -- 4:30Ping Zhong

### Location

During the current year our usual building is being refurbished! I have booked the temporary school seminar room, still called the MALL. This is located in the "Physics Research Deck". On this Map look for the purple building, number 69. I hope to have signs up to direct people: there is one entrance at "71" on the map; an alternative entrance is to get into the long thin building "73" (orange on the map) and then take staircase 4 to level 8. Once inside the physics research deck, you need to go up one floor-- there are large Orange posters directing you. The Seminar room is next to the open-plan common room.

My (Matt's) personal office is currently the other end of campus, in Engineering, but I will locate myself in the common room for the duration of the meeting.

Some General travel information to the School of Maths is avaliable, and some travel information to the university. In particular, there is a Campus Map, also in PDF Format.

### 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).

• There are one or two cafe bars on Blenheim Terrace, building 64.
• Also some pubs and takeaways further up Woodhouse Lane, opposite building 50. And a little Italian restaurant around the corner opposite building 48.
• There are a few cafes and a pub very near the accommodation, on the corner of Springfield Mount and Mount Preston Street, near where buildings 88 and 90 almost meet.
• Lots and lots of things into the city (to the South of the university).

On Thursday 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 Thursday evening, accommodating circa 15 people will not be too hard).

### Accommodation options

For speakers, I hope to arrange accomodation.

For non-speakers, some cheap but good accommodation options, all in the region of £50 a night if you book early:

• Storm Jameson Court - University accommodation on campus; somewhat better than your average halls of residence! I have currently block booked this, with the aim of prioritising access for speakers.
• 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

#### Gabor Elek

##### Operator Algebras over finite fields

One of the most important conjectures in the theory of operator algebras is the Connes Embedding Problem. In this talk, I will introduce the ring theoretic version of the CEP. We shall see, how the analytic notions of amenability, quasidiagonality, almost finiteness, von Neumann rank function appear in the purely algebraic context and how these notions are related to some fundamental open problems of noncommutative ring theory.

#### Marcel de Jeu

##### Positive representations

Many spaces in analysis are ordered (real) Banach spaces, or even Banach lattices, with groups acting as positive operators on them. One can even argue that such positive representations of groups are not less natural than unitary representations in Hilbert spaces, but contrary to the latter they have hardly been studied. The same holds for representations of ordered Banach algebras on ordered Banach spaces where a positive element acts as a positive operator. Whereas there is an elaborate theory of $$^\ast$$-representations of $$C^*$$-algebras, hardly anything is known about positive representations of ordered Banach algebras.

We will sketch the field of positive representations'', and mention some of the main problems (of which there are many) and results (of which there are still few), jointly obtained with Sjoerd Dirksen, Miek Messerschmidt, Ben de Pagter, Bj\"orn de Rijk, Mark Roelands, Jan Rozendaal, Frejanne Ruoff, and Marten Wortel.

The talk is meant as an advertisement for the topic and, more generally, for studying groups and algebras of operators on ordered Banach spaces. The step from single operator theory on Hilbert spaces to groups and algebras of operators was taken in the first half of the 20th century, and now the field of Positivity seems ripe for a similar development.

##### The real and complex technique in harmonic analysis

This talk reviews complex and real techniques in harmonic analysis. We describe the common source of both approaches rooted in the covariant transform generated by the affine group.

#### Derek Kitson

##### Geometric rigidity in normed spaces

The rigidity operator is a fundamental tool for studying the infinitesimal rigidity properties of Euclidean bar-joint frameworks. In this talk we consider a rigidity operator for bar-joint frameworks in a general finite dimensional real normed linear space. Using this operator, we derive necessary Maxwell-Laman-type counting conditions for isostatic bar-joint frameworks with non-trivial symmetry group. These conditions imply very simply stated restrictions on the number of those structural components that are fixed by the various symmetry operations of the framework. In the last part of the talk a new spanning tree characterisation will be presented for two dimensional isostatic frameworks with either reflectional or half-turn rotational symmetry operations in the case of a polyhedral norm with unit ball a quadrilateral. This is joint work with Bernd Schulze.

#### Paul Mitchener

##### Categories of Unbounded Operators

In this talk, I will define a new algebraic structure modelled on the properties of unbounded linear operators between Hilbert spaces, and demonstrate how a version of the Gelfand-Naimark theorem holds.

#### Przemysław Ohrysko

##### On the spectra of measures

In this talk I would like to present the most important topics from the article 'On the relationships between Fourier - Stieltjes coefficients and spectra of measures' (joint work with Micha Wojciechowski) which was published in Studia Mathematica. This work concentrates on problems related to Banach algebra $$M(\mathbb{T})$$ (convolution algebra of measures on the circle group) and especially on spectra of measures. Despite the fact that they can be very complicated in general, we have proved that it is possible to force measures to have natural' spectra (equal to the closure of the range of its Fourier - Stieltjes transform) in a way which uses only whole range of the Fourier - Stieltjes transform without any information on the distribution. Moreover, I will show some recent results on the topological properties of the Gelfand space of the algebra of measures which have not been published yet. In particular, the first proof of non - separability of this space will be given. Additionally, some information about the set of measures which perturb the natural spectrum to the natural spectrum will be provided.

#### Alexey Popov

##### Commutative, amenable operator algebras are similar to C*-algebras

Amenability of Banach algebras was introduced B. Johnson in 1972 and has been a subject of intense study since then. For C*-algebras, this notion coincides with the concept of nuclearity. It was a long-standing question if every nonselfadjoint amenable operator algebra is isomorphic to a C*-algebra. This question was answered in the negative by Choi, Farah, and Ozawa in 2013. Nevertheless, there is a substantial number of partial positive results, and there are very natural classes of operator algebras for which the question is open. For example, even for singly generated algebras only partial answers had been known. In this talk, we will show that the question has affirmative answer for all commutative operator algebras. This is a joint work with L.W. Marcoux.

#### Jan Rozendaal

##### Operator Lipschitz functions on Banach spaces

Operator Lipschitz estimates of the form \begin{align*} \newcommand {\La}{{\mathcal{L}}} \|f(B)-f(A)\|_{\mathcal{L}(H)}\leq C \|B-A\|_{\mathcal{L}(H)}, \end{align*} for normal operators $$A$$ and $$B$$ on a Hilbert space $$H$$, for example $$\ell_{2}$$, have been extensively studied, in particular for $$f$$ the absolute value function. In this talk, we will consider the more general estimate of the form \begin{align*} \|f(B)S-Sf(A)\|_{\mathcal{L}(\ell_{p},\ell_{q})}\leq C \|BS-SA\|_{\mathcal{L}(\ell_{p},\ell_{q})} \end{align*} for so-called diagonalizable operators $$A\in\La(\ell_{p})$$, $$B\in\La(\ell_{q})$$ and $$S\in\La(\ell_{p},\ell_{q})$$, where $$p,q\in[1,\infty]$$. The results that will be presented imply Lipschitz estimates for diagonalizable matrices with a constant independent of the size of the matrix.

The talk is based on joint work with Fedor Sukochev and Anna Tomskova from the University of New South Wales.

#### Richard Skillicorn

##### Extensions and Pullbacks of Banach Algebras

An extension of a Banach algebra $$B$$ is a short exact sequence $$0\longrightarrow I \longrightarrow A \longrightarrow B\longrightarrow 0$$ of Banach algebras and continuous algebra homomorphisms. An extension splits algebraically if we can split $$A$$ into two, in the sense that there is a subalgebra $$C$$ of $$A$$ such that $$A=I\oplus C$$. Further, we say an extension splits strongly if we can find a closed subalgebra $$D$$ of $$A$$ such that $$A=I\oplus D$$. Of course a strong splitting implies an algebraic one. We examine the question of when an algebraic splitting is enough to ensure a strong splitting, or more accurately, for a given Banach algebra $$B$$, is it true that every extension of B which splits algebraically also splits strongly? This leads, in certain cases, to a consideration of pullbacks in the category of Banach algebras and continuous algebra homomorphisms. Our main result shows that for the Banach algebra of bounded operators on a Banach space, and for the Calkin algebra, the answer we get depends on the Banach space in question.

#### Simeng Wang

##### $$L_{p}$$-improving convolution operators on finite quantum groups

For $$A$$ being a finite dimensional $$C^*$$-algebra equipped with a faithful tracial state $$\tau$$, and $$T:A\to A$$ being a unital trace preserving map on $$A$$, we prove that the $$L_{p}$$-improving property $$\|T:L_{p}(A)\to L_{2}(A)\|=1$$ with some $$1<p<2$$ holds if and only if we have the `spectral gap'': $$\sup_{x\in A\backslash\{0\},\tau(x)=0 }\|Tx\|_{2}/\|x\|_{2}<1.$$ As a result we characterize positive convolution operators on a finite quantum group $$\mathbb{G}$$ which are $$L_{p}$$-improving. More precisely, it is proved that the convolution operator $$T_{\varphi}:x\mapsto\varphi\star x$$ given by a state $$\varphi$$ on $$C(\mathbb{G})$$ satisfies $\exists 1<p<2,\quad\|T_{\varphi}:L_{p}(\mathbb{G})\to L_{2}(\mathbb{G})\|=1$ if and only if the Fourier series $$\hat{\varphi}$$ satisfies $$\|\hat{\varphi}(\alpha)\| \leq 1$$ for all nontrivial irreducible unitary representation $$\alpha$$, if and only if the state $$(\varphi\circ S)\star\varphi$$ is non-degenerate (where $$S$$ is the antipode). We also prove that these $$L_{p}$$-improving properties are stable under taking free products, which gives a method to construct $$L_{p}$$-improving multipliers on infinite compact quantum groups.

#### Mateusz Wasilewski

##### Non-maximal subspaces of maximal operator spaces

Every Banach space admits many different operator space structures; we will concentrate on the maximal and minimal ones, which are related to each other by duality. The class of maximal operator spaces is closed under quotients but, in general, not under passing to subspaces. We will exhibit many examples of subspaces of maximal operator spaces that are not maximal. By duality, we will obtain non-minimal quotients of minimal operator spaces. As a by-product, we will be able to answer the question of Vern Paulsen concerning amalgamated direct sums of minimal operator spaces.

#### Ping Zhong

##### Superconvergence to freely infinitely divisible laws

Given an infinitely divisible distribution $$\nu$$ relative to free independence in the sense of Voiculescu, let $\mu_n$ be a sequence of probability measures and let $$k_n$$ be an increasing sequence of integers such that $$(\mu_n)^{\boxplus k_n}$$ converges weakly to $$\nu$$. We show that the density $$d(\mu_n)^{\boxplus k_n}/dx$$ converges uniformly to the density of $$d\nu/dx$$ except possibly in the neighborhood of one point. This phenomenon is called superconvergence. We also obtain corresponding results for free multiplicative convolution. The results are joint with Michael Anshelevich, Hari Bercovici and Jiun-Chau Wang.

## Participants

2. Amerah Alameer     Leeds
3. Amer Albargi     Leeds
4. Aolo Bashar Abusaksaka     Leeds
5. Alexander Belton     Lancaster
7. Yemon Choi     Lancaster
8. Garth Dales     Lancaster
9. Matthew Daws     Leeds
10. Marcel de Jeu     Leiden
11. Gabor Elek     Lancaster
12. Sian Fryer     Leeds
13. Jason Hancox     Lancaster
14. Robin Hillier     Lancaster
15. Mateusz Jurczyński     Lancaster
17. Derek Kitson     Lancaster
18. Andrzej Kucik     Leeds
19. Niels Laustsen     Lancaster
20. Martin Lindsay     Lancaster
21. Paul Mitchener     Sheffield
22. Harish Reddy Mulakkayala     Lancaster
23. Khawlah Mustafa     Leeds
24. Przemysław Ohrysko     IMPAN
25. Alexey Popov     Newcastle
26. Jan Rozendaal     TU Delft
27. David Salinger     Leeds
28. Richard Skillicorn     Lancaster
29. Steve Trotter     Leeds
30. Simeng Wang     Besançon and Warsaw
31. Mateusz Wasilewski     Warsaw
32. Jared White     Lancaster
33. Richard Whyman     Leeds
34. Ping Zhong     Lancaster