## 50th Anniversary meeting of the North British Functional Analysis Seminar

12-14 April 2018

ICMS, 15 South College Street, Edinburgh, EH8 9AA

The North British Functional Analysis Seminar (NBFAS) was founded in February 1968 by the late Frank Bonsall and John Ringrose with a meeting at the University of Edinburgh. The founding universities were Edinburgh and Newcastle. Since then, almost 200 meetings took place at the growing number of member institutions (at present 14 universities in England, Scotland and Northern Ireland) with the purpose to bring to the UK leading figures in functional analysis that are based outside the UK.

In 2018, NBFAS will turn 50 years old. An afternoon of the current meeting will be dedicated to a review of the historical development of NBFAS including contributions by seasoned members. Professor Ringrose has agreed to participate in this session.

Speakers include:
Nate Brown, Penn State
Michel Ledoux, Toulouse
Nicolas Monod, EPFL Lausanne
Assaf Naor, Princeton
Jonathan M. Rosenberg, Maryland
Stuart White, Glasgow
(see below for abstracts)

The workshop is grateful for financial support from:

## Arrangements

### Venues

The workshop will take place at ICMS, 15 South College Street, Edinburgh, EH8 9AA on Thursday 12 April and Friday 13 April.

On Saturday 14 April, it will be held in Lecture Theatre A, David Hume Tower, George Square, Edinburgh EH8 9JX.

### Registration

Confirmed participants should aim to arrive at ICMS between 12:15 and 13:00 on Thursday 12 April. Registration will take place in the Chapterhouse on level 1 (ground floor).

#### PLEASE NOTE FOR THE REGISTRATION FEE OF £20 (£10 ECR’S) AND THE DINNER FEE OF £35, WE WILL ONLY ACCEPT CASH PAYMENTS.

The workshop will commence at 13:00 with the first talk in the Newhaven Lecture Theatre on Level 4. Details of the Preliminary Schedule can be found below.

### Workshop dinner

A workshop dinner will take place on Friday 13 April at 18.30 at Blonde, St Leonard’s Street. There is a non-refundable contribution of 35GBP towards the cost of this dinner. This will be payable on arrival at ICMS. Places for dinner are limited due to the capacity of the venue. If you did not indicate that you wanted to attend on the application form but have changed your mind then you may request one of the remaining places at registration. These will be allocated on a first come, first served basis.

### Catering

• Tea/coffee will be provided in the Chapterhouse, each day, mid-morning and afternoon;
• Lunch will be provided in the Chapterhouse on Friday only;
• Tea/coffee will be provided at the morning break. The workshop will close around lunchtime on Saturday. Please note: no lunch will be provided.

### Financial support

Requests for financial support are now closed. Limited funding was available to assist attendance by particularly early career researchers based in the UK. If your request was successful the support will take the form of the cost of advance purchase rail fares and up to 60 GBP (early career: 40 GBP) towards your accommodation expenses per night. You will be given a claim form at registration.

### Call for short talks

Applications from early career researchers for a session of contributed 30 minute talks on Thursday afternoon are now closed. Titles and abstracts can be seen below.

### UK visas

If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site [https://www.gov.uk/apply-uk-visa] to find out if you need a visa and if so how to apply for one.

### Travel

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies that may occur on your trip.

A taxi directly from the airport will cost approximately 20.00 to 25.00 GBP to the city centre for a one-way journey. There is also a bus service direct from the airport to the city centre which will cost 4.50 GBP single or 7.50 GBP return - the Airlink 100. This is a frequent service (every 10 minutes during peak times) and will bring you close to Waverley Railway Station, only a short walk to the accommodation and the workshop venue (see map in 'Venue' section above).

Lothian buses charge £1.70 for a single, £4.00 for a day ticket. Please note that the exact fare is required and no change is given.

If travelling by train, please note that Edinburgh has several railway stations - Waverley Railway Station being the main station and closest to the workshop venue at 15 South College Street. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge.

### Accommodation in Edinburgh

Some suggestions for accommodation close to the workshop venue.

• Ibis Hotel on South Bridge (not Hunter Square) Very close to ICMS.
• Motel One Edinburgh (Please note that Motel One has two hotels in the city centre - Motel One Royal or Motel One Princes.  Both are within easy walking distance of ICMS)
• Pollock Halls of Residence, The University of Edinburgh, 18 Holyrood Park Road, Edinburgh EH16 5AY 0800 028 7118 (UK only) - +44 (0)131 651 2189 - University of Edinburgh accommodation. Pollock Halls is about a 20-25 minute walk.
• Jurys Inn Edinburgh, 43 Jeffrey Street, Edinburgh EH1 1DH +44 (0)131 200 3300 - A 5 minute walk from ICMS.
• Edinburgh City Hotel, 79 Lauriston Place, Edinburgh, EH3 9HZ  +44 (0)131 622 7979 -
• St Christophers Hostel, 9-13 Market Street, Edinburgh EH1 1DE  (approx 25.00 or less depending on offers available)+44 (0)20 7407 1856 - bookings@st-christophers.co.uk This is Youth Hostel accommodation.

### Talks and audio/visual equipment (for invited speakers)

Talks will be 50 minutes in length with 10 minutes for questions and answers. All talks will be held in the Newhaven Lecture Theatre. The lecture theatre has a built in computer, data projector, and visualiser/document camera. In addition, there are two blackboards. The projector and one board may be used simultaneously. We advise speakers that, where possible, you bring your talk on a memory stick/USB to put onto our computer in advance of your session - either at the start of the day or during coffee/lunch breaks. ICMS staff can help with this. It is possible for you to use your own laptops but it is then your own responsibility to ensure that resolutions are changed on the laptops if necessary (ICMS will not change the resolution on our equipment). If you use a Mac we expect you to bring your own adaptor.

## Preliminary schedule

Thursday
13:00 - 17.30 Early career talks
18.00 NBFAS committee meeting

Friday
9:30 - 10.30 Nate Brown, Penn State University
10:30 - 11.00 coffee break
11.00 - 12.00 Michel Ledoux, University of Toulouse
12:00 - 13.00 Jonathan Rosenberg, University of Maryland
13.00 - 14.00 Lunch
14.00 - 15.00 Nicolas Monod, Ecole Polytechnique Federale de Lausanne
15.00 - 16.00 John Ringrose, Alastair Gillespie
16.00 - 16.30 coffee break
16.30 - 18.00 E. Chris Lance, Allan Sinclair, greetings
18:30 conference dinner

Saturday
9:30 - 10.30 Assaf Naor, Princeton University
10.30-11.00 coffee break
11.00 - 12.00 Stuart White, University of Glasgow

## Contributed talks

Contributed talks on Thursday will be organised in two parallel sessions of 30-minute talks in the Newhaven Lecture Theatre, and the Seminar Room

### Session 1

13:00-17:30 (Newhaven Lecture Theatre, Level 4)
Uwe Franz, University of Bourgogne Franche-Comte
James Gabe, University of Glasgow
Joachim Moussounda, Blessington Christian University
Daniel Virosztek, Institute of Science and Technology
Maxim Zyskin, University of Nottingham
Moritz   Weber, Saarland University
Yong Zhang, University of Manitoba

### Session 2

13:00-17:30 (Seminar Room, Chapterhouse, Level 1)
Aldric Brown, University College London
Tony Carbery, University of Edinburgh
Alexander Helemskii, Moscow State (Lomonosov)University
Bence Horvath, Lancaster University
Xingni Jiang, Mathematical Institute, Leiden University
Houray Melkonian, Heriot-Watt University
Eskil Rydhe, University of Leeds
Sven- Ake Wenger, Teeside University

## Plenary speakers' abstracts

Nathaniel Brown
Simple C*-algebras and topological dimension
In Fall 2015 a capstone theorem was proved, nearly completing a 40-year project. The result is analogous to the classification of finite simple groups, simple Lie groups or various classes of manifolds (e.g. compact surfaces, hyperbolic manifolds), but this time the objects are simple C*-algebras with finite topological dimension. In this talk I will start at the beginning, with lots of examples, and discuss a few of the highlights that led to the classification theorem.

Michel Ledoux
How does the heat equation explore geometric and functional inequalities

As is classical, the heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time. As the prototypical parabolic partial differential equation and via its connection with Brownian motion in probability theory, the heat equation is of fundamental importance in diverse scientific fields, including (Riemannian) geometry, Lie groups, mathematical physics, graph theory, up to mechanics or biology.

During the last decades, the gift of ubiquity of the heat equation has developed to approach various geometric and functional inequalities. Starting from the elementary example of Hölder’s inequality, the talk will feature illustrations of the heat flow method to families of integral inequalities, Sobolev inequalities and more refined isoperimetric theorems in Euclidean and Riemannian spaces. The method developed recently in discrete Boolean analysis motivated by questions in theoretical computer science.

Nicolas Monod
Asymptotics of isoperimetry and unitarisable representations
I will review the Dixmier Problem, open since 1950, which asks for a condition ensuring that all representations of a given group are unitarisable. This can be related to analytic conditions and to geometric constructions on groups.
I will also present new work with Gerasimova, Gruber and Thom, where the Dixmier Problem is related to the asymptotic properties of isoperimetric constants.

Assaf Naor
Vertical perimeter versus horizontal perimeter
We will show that the appropriately-defined vertical perimeter of a measurable subset of the Heisenberg group is at most a constant multiple of its horizontal (Heisenberg) perimeter. This isoperimetric-type inequality exhibits different behavior in dimension 3 and 5, and it has a variety of applications to several areas.
Joint work with Robert Young.

Jonathan Rosenberg
Index theory on manifolds with singularities, C*-algebras, and positive scalar curvature
The philosophy of noncommutative geometry suggests that noncommutative C*-algebras should be a useful tool in analysis on singular spaces. We illustrate this in the case of index theory on certain manifolds with singularities, and apply it to characterising when such singular manifolds admit metrics of positive scalar curvature. This is in part joint work with Boris Botvinnik and Paolo Piazza.

Stuart White
Quasidiagonality vs Amenability
Quasidiagonality was introduced by Halmos as approximation property for operators, and has subsequently been studied in the setting of families of operators, where it has a somewhat mysterious topological nature. I’ll survey quasidiagonality, focusing on reduced group C*-algebras, where Rosenberg noticed that quasidiagonality of the reduced C*-algebra of a group entails amenability of the underlying group, and conjectured the converse. I’ll discuss the solution to this conjecture, and how the ideas contribute to the classification of simple amenable C*-algebras. This is based on joint work with Aaron Tikuisis and Wilhelm Winter.

## Abstracts for contributed talks

Aldric Brown
On lower semi-continuity of metric projections onto finite dimensional subspaces of continuous function spaces.
Let T denote a locally compact Hausdorff space, C(T) the space of continuous real valued functions on T that tend to zero at infinity, and let G be a finite dimensional subspace of C(T), of dimension n. Then PsubG denotes the metric projection of C(T) onto G. Approximation theorists are interested in properties of PsubG, in particular (1) Is P(f) a single function for each f in C(T) ? (in which case P is continuous and one says that G is a Chebyshev subspace); (2) if G is not Chebyshev then is PsubG a lower semi-continuous set valued mappring ? (3) Is there a continuous function r: C(T) --> G, a 'continuous selection', such that, for each f in C(T) , r(f) is a point of PsubG(f) ? Obviously, and with an obvious meaning', (1) implies (2) implies (3) . The talk will present a complete and precise determination of those P and G such that Psub G is lower semi-continuous. The question is approximation theoretic, the result is functional analytic.

Tony Carbery
A multilinear Maurey factorisation theorem
A famous theorem of B. Maurey states that if T is a positive linear map from a normed lattice Y to the Borel measurable functions on a locally finite Hausdorff topological space X which is bounded from Y into L^q(X) with 0 < q < 1, then T factorises through L^1 -- that is, there exists a suitable probability density w such that T is bounded from Y into L^1(w^(q-1)/q). In joint work with Stefan Valdimarsson we give a multilinear analogue of this theorem which has surprising (or perhaps unsurprising?) links to harmonic analysis and the study of Kakeya sets in particular.

Uwe Franz
Invariant Markov semigroups on quantum homogeneous spaces
Invariance properties of functionals and linear maps on algebras of functions on quantum homogeneous spaces are studies, in particular for the special case of expected coideal *-subalgebras. Several one-to-one correspondences between such invariant functionals and linear maps acting on function algebras associated to the quantum homogeneous space and those associated to the compact quantum underlying the action are established. Applying these results to convolutions semigroups of states and to quantum Markov semigroups, we show how quantum Markov semigroups on an expected coideal *-subalgebra can be classified. The classical spheres $S^{N-1}$, the free spheres $S^{N-1}_+$, and the half-liberated spheres $S^{N-1}_*$ are considered as examples. Joint work with Biswarup Das and Xumin Wang.

James Gabe
Classification of purely infinite C*-algebras
I will present the general strategy for classifying separable, nuclear, purely infinite C*-algebras.

Alexander Helemskii
Projectivity, freeness and tensor products in matricially normed spaces
I shall report about several results, concerning general (not necessarily operator) matricially normed spaces in the sense of Effros/Ruan. First, we describe metrically projective and metrically free matricially normed spaces in terms of a special space $\;widehat M_n$, the space of $n\;times n$ matrices, endowed with a special matrix--norm. We show that metrically free matricially normed spaces are matricial $\;ell_1$--sums of some distinguished families of matricially normed spaces $\;widehat M_n$, whereas metrically projective matricially normed spaces are complete direct summands of matricial $\;ell_1$--sums of arbitrary families of spaces $\;widehat M_n$. Second, we show that matricially normed spaces have a special tensor product possessing the universal property with respect to completely bounded bilinear operators. We study some general properties of this tensor product (among them a kind of adjoint associativity), and compute it for some tensor factors, notably for $L_1$ spaces. In particular, we obtain what could be called the matricially normed version of the Grothendieck theorem about classical projective tensor products by $L_1$ spaces.

Bence Horvath
Perturbations of homomorphisms between Banach algebras
Perturbations of characters of Banach algebras were first studied by Jarosz in [1]. His work was substantially extended by Johnson in [2] and [3], where the target space C (complex numbers) is replaced with an arbitrary Banach algebra B. In this latter paper Johnson defines the so-called AMNM property (Almost Multiplicative implies Near Multiplicative) for a pair of Banach algebras (A, B). Roughly speaking, this property is concerned with the following question: Let A, B be Banach algebras. For a bounded linear map φ : A → B let us define the multiplicative defect of φ as def (φ) := sup{ \;Vert φ(ab) − φ(a)φ(b) \;Vert : a, b ∈ A, \;Vert a \;Vert , \;Vert b \;Vert ≤ 1}. Let Mult(A, B) denote the (closed) set of continuous algebra homomorphisms from A to B. For a bounded linear map φ : A → B, how does the quantity def (φ) relate to dist(φ, Mult(A, B))? In our talk we shall investigate this question with special emphasis on algebras of operators on Banach spaces. This question is very intimately related to amenability and bounded Hochschild cohomology of Banach algebras. Time permitting we will present a result extending the main result of Johnson in [3] which also allows us to prove that (B(X),B(X)) is AMNM for a large class of non-Hilbertian Banach spaces X.
Joint work with Y. Choi and N.J. Laustsen.
[1] Krzysztof Jarosz. Perturbations of Banach algebras, Springer-Verlag, Berlin,
[2] B.E. Johnson. Approximately multiplicative functionals, J. London Math. Soc, 1986.
[3] B.E. Johnson. Approximately multiplicative maps between Banach algebras, J. London Math. Soc, 1988.

Xingni Jiang
Positive representations of C0(X)
If $X$ is a locally compact Hausdorff space, then a representation of the complex $\mathrm{C}^\ast\!$-algebra $\mathrm C_0(X)$ on a Hilbert space $H$ is given by a spectral measure that takes its values in the orthogonal projections on $H$. It is natural to ask whether something similar is true for a positive representation of the ordered Banach algebra $\mathrm{C}_0(X)$ on a Banach lattice $E$. If $E$ is a KB-space (e.g\ if $E$ is an ${\mathrm L}^p$-space for finite $p$, or if $E$ is reflexive), then the answer is affirmative: the representation is given by a spectral measure that takes its values in the positive projections on $X$; see [1].

The proofs in [1] make use of the fact that $E$ is a Banach space, but some results in [1] suggest that a purely order-theoretic more general approach might also be possible. In this lecture, we shall explain that this is indeed the case.

As a preparation, we shall sketch an integration theory for measures taking values in a suitable partially ordered vector space $E$. After that, we shall discuss a Riesz representation theorem for a positive map $T\colon\mathrm{C}_0(X)\to E$. Under mild conditions, this is given by a positive $E$-valued measure. In the next step, we apply the previous result to a positive representation $\pi\colon\mathrm{C}_0(X)\to A$, where $A$ is a suitable partially ordered algebra. In that case, the pertinent positive $A$-valued measure takes values in the idempotents of $A$.

If $A$ equals the regular operators on a suitable partially ordered vector space $E$, then the previous result yields a spectral measure for $\pi$ that takes its values in the positive projections on $E$. This result has not only the main result in [1] as a special case, but also the aforementioned existence of spectral measures for representations of $\mathrm C_0(X)$ on a Hilbert space.

In the end, we discuss which positive representation could automatically be a lattice homomorphism.

This is joint work with Marcel de Jeu.

[1] M. de Jeu, F. Ruoff, Positive representations of $\mathrm{C}_0(X).$ $\!\!$I, Ann. Funct. Anal. (2016), 180--205.

Using entropy to parametrize KMS-states of Pimsner algebras
We revisit the Laca-Neshveyev classification of KMS-states at positive inverse temperature for Pimsner algebras of finite rank. The finite rank entails an entropy notion that shapes the KMS simplices and allows to parametrize the finite and the infinite parts by tracial states on the diagonal. In particular the tracial entropies dictate the lowest critical temperature below which there are no KMS-states and the strong entropy is the maximum above which there are no infinite KMS-states. As an application we recover previous results for graph C*-algebras and generalized C*-crossed products.

Houry Melkonian
Riesz basis criteria for families of dilated periodic functions
Consider a periodic function f, such that its restriction to the unit segment lies in the Banach space L^2 = L2(0; 1). Denote by S the family of dilations f(nx) for all n positive integer. The purpose of this talk is to discuss the following question: When does S form a Riesz basis of L^2? In this talk, we will present a new mutli-term criteria for determining Riesz basis properties of S in L^2. This method was established in [L. Boulton, H. Melkonian; arXiv: 1708.08545 J. (2017)] and it relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give suffcient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coeffcients. We will then examine the application of these criteria in the case of f being the p-trigonometric functions. These functions arise naturally in the context of the non-linear eigenvalue problem associated to the one-dimensional p-Laplacian in the unit segment. These results improve upon those of [D. E. Edmunds, P. Gurka, J. Lang, J. Math. Anal. Appl. 420 (2014)] and [L. Boulton, H. Melkonian J. Math. Anal. Appl., 444 (2016)].

Joachim Moussounda
On von Neumann’s Inequality for upper (or lower) Complex Triangular Toeplitz Contractions
We prove that von Neumann’s inequality holds for tuples of contractions which are embedded in tuples of circulant contractions.

Eskil Rydhe
Cyclic m-isometries, and Dirichlet type spaces
We consider cyclic m-isometries on a complex separable Hilbert space. Such operators are characterized in terms of shifts on abstract spaces of weighted Dirichlet type. Our results resemble those of Agler and Stankus, but our model spaces are described in terms of Dirichlet integrals rather than analytic Dirichlet operators. The chosen point of view allows us to construct a variety of examples.

Daniel Virosztek
Characterizations of central elements by local monotonicity and local convexity of certain functions on C^*-algebras
Connections between algebraic properties of C^*-algebras and some fundamental properties of functions defined on them by functional calculus have been investigated widely. Some of the classical results characterize the commutativity of a C^*-algebra by the global monotonicity of certain functions. For example, Ogasawara's theorem says that a C^*-algebra is commutative if and only if the square function is monotone increasing (with respect to the partial order induced by positivity) on its positive cone [2]. Other results establish connections between the global convexity of some functions and the commutativity of the C^*-algebra (see, e.g., [3]). The aim of our talk is to "localize" some of the global results, that is, to give characterizations of the central elements of the C^*-algebra in terms of local monotonicity and local convexity of certain functions. We introduce two quite wide function classes and show that each element of these function classes distinguishes central elements and non-central ones by local monotonicity or local convexity. We remark that the idea of considering local versions of classical results comes from Molnar [1]. Also, we point out how local statements imply their global counterparts immediately. Our talk is based on the papers [5] and [4].
[1] L. Molnár, A characterization of central elements in C^*-algebras, Bull. Austral. Math. Soc. 95 (2017), 138–143.
[2] T. Ogasawara, A theorem on operator algebras, J. Sci. Hiroshima Univ. Ser. A. 18 (1955), 307–309.
[3] S. Silvestrov, H. Osaka and J. Tomiyama, Operator convex functions over C^*-algebras, Proc. Eston. Acad. Sci. 59 (2010), 48–52.
[4] D. Virosztek, Characterizations of centrality by local convexity of certain functions on C^*-algebras, to appear in Operator Theory: Advances and Applications in 2018. Available online: https://arxiv.org/abs/1709.03357
[5] D. Virosztek, Connections between centrality and local monotonicity of certain functions on C^*-algebras, J. Math. Anal. Appl. 453 (2017), 221–226.

Moritz Weber
Quantum groups by example
We will approach compact quantum groups in a pedestrian paste meeting good old friends such as C*-algebras, Gelfand-Naimark’s Theorem and K-Theory on our way. We finally end up with a brief survey regarding links between the combinatorics of set partitions, C*-algebras and so called easy quantum groups.

Sven- Ake Wenger
The heart of the Banach spaces
Let X be a Banach space and let Y be a linear subspace. If Y is closed in X then X/Y is a Banach space in the quotient norm. If Y is not closed then this wrong---even if Y is a Banach space in a norm stronger than those induced by X. A prominent example is Y=l^1 and X=c_0 The unpleasant fact, that there is no reasonable Banach space X/Y in the setting above, motivated Waelbroeck in the 1960s to consider formal quotients of Banach spaces. Amazingly, in 1982, the same year in which he published his paper on the category of quotient Banach spaces, Beilinson, Bernstein and Deligne published in a geometric context a very abstract and by now very famous theory about hearts of t-structures on triangulated categories. It turnes out that in their terminology, and for the special case of Banach spaces, the heart is precisely the category of formal quotients considered by Waelbroeck. In the talk we sketch the definition of the heart and discuss possible generalizations beyond the case of Banach spaces.

Yong Zhang
Amenability and fixed points of semigroup actions Ab
We investigate common attractive points of nonexpansive semigroup action on a subset of a Banach space. If $E$ is a strictly convex and reflexive Banach space and $S$ is a semigroup acting on a closed convex subset $C$ of $E$, we show that the existence of a common attractive point implies the existence of a common fixed point for the action. For a nonexpansive semigroup action on a subset of a Hilbert space, we show that the existence of a common attractive point is ensured by various amenability properties of the semigroup. This is joint work with A. T.-M. Lau.

Maxim Zyskin
Variational problems in continuum mechanics of defective crystals
In Davini-Parry approach, continuum mechanics of defective crystals is described in terms of continuum frame of 'lattice vector fields', dislocation density tensor encoding information on Lie bracket of those vector fields, and appropriate number of higher order derivatives of dislocation density tensor. The main constituitive assumption is that the Lie algebra of lattice vector fields is finite dimensional. I will present examples of such defective crystal structures, and discuss Young measure valued minimizers of corresponding variational problems.