## SCIENTIFIC PROGRAMS AND ACTIVITIES

October 31, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
 January-June 2014 Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras May 2014 Theme Period on Operator Spaces, Locally Compact Quantum Groups and Amenability Organizer: Volker Runde
 May 12-16, 2014 Mini-Courses May 20-23, 2014 Mini-Courses May 26-30, 2014 Workshop on Operator Spaces, Locally Compact Quantum Groups and Amenability Back to main index

May 12-23, 2014
Mini-Courses: Operator Spaces, Locally Compact Quantum Groups and Amenability

Video archive of talks

 Monday May 12 Room 230 Tuesday May 13 Room 230 Wednesday May 14 Room 230 Thursday May 15 Room 230 Friday May 16 Room 230 10:00 a.m. Volker Runde Amenability of Banach algebras,1 Volker Runde Amenability of Banach algebras,3 Michael White Cohomology of Banach and topological algebras,1 (lecture notes) Michael White Cohomology of Banach and topological algebras,3 (lecture notes) 11:00 a.m. David Blecher, Operator spaces,1 (lecture notes) David Blecher, Operator spaces,3 (lecture notes) Piotr Soltan On amenability and injectivity for locally compact quantum groups Lunch Break 2:00 p.m. Volker Runde Amenability of Banach algebras,2 Volker Runde Amenability of Banach algebras,4 Michael White Cohomology of Banach and topological algebras.2 Michael White Cohomology of Banach and topological algebras,4 3:30 p.m. David Blecher Operator spaces,2 David Blecher Operator spaces,4
Piotr Soltan
On amenability and injectivity for locally compact quantum groups

I will recall the notion of amenability of a locally compact quantum group and prove a theorem providing a characterization of this property in terms of injectivity of the von Neumann algebra associated to the dual (quantum) group.

Minicourses Week of May 20-23

 Monday May 19 Tuesday May 20 Room 230 Wednesday May 21 Room 230 Thursday May 22 Stewart Library Friday May 23 Stewart Library 10:00 a.m. Victoria Day Holiday Matt Daws Locally Compact Quantum Groups,1 (lecture notes) Matt Daws Locally Compact Quantum Groups (lecture notes) 11:00 a.m. Zinaida Lykova Higher-dimensional amenability,1 (lecture notes) Zinaida Lykova Higher-dimensional amenability (lecture notes) Ebrahim Samei Quantization of topological homology Ebrahim Samei Quantization of topological homology Lunch Break 2:00 p.m. Matt Daws Locally Compact Quantum Groups,2 (lecture notes) Matt Daws Locally Compact Quantum Groups (lecture notes) 3:30 p.m. Zinaida Lykova Higher-dimensional amenability,2 Zinaida Lykova Higher-dimensional amenability Ebrahim Samei Quantization of topological homology Ebrahim Samei Quantization of topological homology

### May 26-30, 2014 Workshop on Operator Spaces, Locally Compact Quantum Groups and Amenability

 Monday Tuesday Wednesday Thursday Friday 10:00-10:50 Leonid Vainerman Coamenability and quantum groupoids . Mahmood Alaghmandan Hypergroups and their amenability notions Ngai-Ching Wong Isometries of real Hilbert C*-modules Rui Okayasu Haagerup approximation property and positive cones associated with a von Neumann algebra Martijn Caspers The Haagerup property for arbitrary von Neumann algebras 10:00-10:50 Coffee Break 11:10-12:00 Reiji Tomatsu Product type actions of compact quantum groups Hun Hee Lee Weighted Fourier algebras on non-compact Lie groups and their spectrumy Yemon Choi Unitarizable group representations and amenable operator algebras Krzysztof Piszczek Amenability properties of the noncommutative Schwartz space Jason Crann An uncertainty principle for unimodular quantum groups 11:10-11:35 Mateusz Jurczynski Quantum Wiener chaos expansion Yuhei Suzuki Amenable minimal Cantor systems of free groups arising from diagonal actions 11:40-12:05 12:10-13:00 Pierre Fima Graphs of quantum groups and K-amenability Michael Brannan $L_p$-representations of discrete quantum groups Yong Zhang Amenability properties of Banach algebra valued continuous functions Mehrdad Kalantar Quantum groups actions and noncommutative boundaries David Kyed Dimension-flatness and Lück's amenability conjecture. 12:10-13:00 Lunch Break 15:30-16:20 Chi-Keung Ng Property T for locally compact quantum groups. Coxeter Lecture Sorin Popa On II1 factors arising from free groups acting on spaces Coxeter Lecture Sorin Popa On II1 factors arising from free groups acting on spaces Coxeter Lecture Sorin Popa On II1 factors arising from free groups acting on spaces Alexei Pirkovskii Taylor’s functional calculus and derived categories 15:30-16:20 Coxeter Lecture Reception Sutanu Roy The Haagerup property of the generalised Drinfel'd double. Gilles Pisier A continuum of $\mathrm{C}^*$-norms on ${{\mathbb B}}(H)\otimes {{\mathbb B}}(H)$ and related tensor products 16:30
Mahmood Alaghmandan
Hypergroups and their amenability notions (slides)
Abstract: In this talk, after defining hypergroups and introducing algebras constructed on them, we discuss their different amenability notions. We specifically consider these notions for some classes of hypergroup structures related to locally compact groups. Subsequently, we demonstrate some applications to locally compact groups and their Banach algebras.
Michael Brannan
$L_p$-representations of discrete quantum groups.
Given a unimodular discrete quantum group $\mathbb G$, we define and study unitary representations of $\mathbb G$ associated to the non-commutative $L_p$-spaces $L_p(\mathbb G)$. After discussing some general aspects of $L_p$-representations, we will show how this theory can be applied to construct new examples of exotic quantum group norms on the algebras of polynomial functions on duals of unimodular orthogonal free quantum groups. This is joint work with Z.-J. Ruan.
Martijn Caspers
The Haagerup property for arbitrary von Neumann algebras (slides)
Abstract: We introduce a natural generalization of the Haagerup property of a finite von Neumann algebra to an arbitrary von Neumann algebra equipped with a normal, semi-finite, faithful weight and prove that this property does not depend on the choice of the weight. In particular this defines the Haagerup property as an intrinsic invariant of the von Neumann algebra. Our initial definition/approach is in terms of cp maps preserving a given weight and therefore stays close to the semi-finite case by M. Choda/P. Jolissaint. However, our techniques rely on crossed product duality.
We shall discuss stability properties of the Haagerup property regarding crossed products and free products. We also show how to define a noncommutative counterpart of the group-Haagerup property in terms of the existence of a proper, continuous, conditionally negative de?nite function.
Our results are motivated by recent examples from the theory of discrete quantum groups, where the Haagerup property appears a priori only with
respect to the Haar state. We will review some of these examples.
This is joint work with Adam Skalski.
Yemon Choi
Unitarizable group representations and amenable operator algebras (slides)

All bounded continuous representations of a locally compact amenable group on a Hilbert space H are unitarizable inside B(H): this is an old result of Dixmier and Day. Is the same true for representations inside the Calkin algebra? It turns out that the answer is sometimes yes and sometimes no, with a crucial role played by the size of the group. In this talk I will explain more precisely what this means, and show how the non-unitarizable representations may be combined with an ingenious idea of Ozawa to produce the first known examples of amenable operator algebras that are not isomorphic to C*-algebras. If time permits I will discuss where the study of amenable operator algebras might go from here. This is based on joint work with I. Farah and N. Ozawa.
Jason Crann
An uncertainty principle for unimodular quantum groups (slides)

Heisenberg's celebrated uncertainty principle, concerning measurements of position and momentum, roughly states that a function and its Fourier transform cannot both be highly concentrated. In 1957, Hirschman extended this uncertainty relation to locally compact Abelian groups by using the relative entropy with respect to the Haar measure to quantify the degree of concentration. In this talk, we will extend Hirschman's result to unimodular quantum groups and discuss various consequences along with partial results in the non-unimodular setting. Time permitting, we will present a potential application to non-commutative random walks on arbitrary discrete quantum groups. This is joint work with Mehrdad Kalantar.
Pierre Fima
Graphs of quantum groups and K-amenability

We construct the fundamental group of a graph of quantum groups and we show that it is K-amenable whenever the initial quantum groups are amenable. This is a joint work with A. Freslon.

Mateusz Jurczynski
Quantum Wiener chaos expansion (slides)
Operator spaces provide a natural framework for investigating quantum stochastic processes. In this talk we will present some important classes of such processes, all of which have the special property of being representable as quantum Wiener integrals". Direct applications include Lie--Trotter product formula for quantum stochastic evolutions and the construction of Levy processes on compact quantum groups.

Quantum groups actions and noncommutative boundaries
We study various properties of noncommutative Poisson boundaries of convolution maps arising from locally compact quantum group actions. We also introduce a quantum version of Jaworski's notion of SAT actions, and investigate basic properties of such actions. The is based on joint work with M. Amini and M. S. M. Moakhar, and M. Kennedy.
David Kyed
Dimension-flatness and Lück's amenability conjecture.
I will discuss the notion of dimension flatness, as well as its connection with amenability, and present a positive answer to a diffuse analogue of Lück's amenability conjecture. The talk is based on joint works with V. Alekseev and H.D. Petersen.
Hun Hee Lee
Weighted Fourier algebras on non-compact Lie groups and their spectrum (slides)
In this talk we will discuss a model for a weighted version of Fourier algebras on non-compact Lie groups. If we recall that the spectrum of the Fourier algebra is nothing but the underlying group itself (as a topological space), then it is natural to be interested in determining the spectrum of weighted algebras. We will demonstrate that the spectrum of the resulting commutative Banach algebra is realized inside the complexification of the underlying Lie group by focusing on the case of Heisenberg group and determine them in some concrete cases.
Chi-Keung Ng
Property T for locally compact quantum groups (slides)
We generalized several equivalences of property T to the case of general locally compact quantum groups. When G is a locally com- pact quantum group of Kac type, we also show that G has property T if and only if every finite dimensional irreducible ¤-representation of Cu 0 (b G) is an isolated point in the spectrum of Cu 0 (b G). On the other hand, as a first step in an attempt to generalize the Delmorme- Guichardet theorem, we introduce, in the case of a second countable locally compact group G, a cohomology theory for the convolution ¤-algebra Cc(G), equipped with an inductive limit topology and the canonical character "G : Cc(G) ! C, and show that the vanishing of all such first cohomologies is equivalent to the property T of G.
[Based on joint works with Chen Xiao]
Rui Okayasu
Haagerup approximation property and positive cones associated with a von Neumann algebra (slides)
We discuss various definitions of the Haagerup approximation property for an arbitrary von Neumann algebra. As a consequence, we give a simple and direct proof that the definition given by M. Caspers and A. Skalski is equivalent to our original one defined by using the standard form. Our strategy is to use the one-parameter family of positive cones due to H. Araki. We also discuss the Haagerup approximation property for non-commutative Lp-spaces. This is based on a joint work with Reiji Tomatsu.
Gilles Pisier
A continuum of $\mathrm{C}^*$-norms on ${{\mathbb B}}(H)\otimes {{\mathbb B}}(H)$\\ and related tensor products (slides) (abstract)
This is an account of joint work with N. Ozawa.
For any pair $M,N$ of von Neumann algebras such that the algebraic tensor product $M\otimes N$ admits more than one $\mathrm{C}^*$-norm, the cardinal of the set of $\mathrm{C}^*$-norms is at least ${2^{\aleph_0}}$. Moreover there is a family with cardinality ${2^{\aleph_0}}$ of injective tensor product functors for $\mathrm{C}^*$-algebras in Kirchberg's sense. Let $\mathbb B=\prod_n M_{n}$. We also show that, for any non-nuclear von Neumann algebra $M\subset \mathbb B(\ell_2)$, the set of $\mathrm{C}^*$-norms on $\mathbb B\otimes M$ has cardinality {\it equal to} $2^{2^{\aleph_0}}$. The talk will also recall the connection of such questions with the non-separability of the set of finite dimensional (actually $3$-dimensional) operator spaces which goes back to a 1995 paper with Marius Junge, and several recent quantitative" refinements obtained using quantum expanders.

Alexei Pirkovskii
Taylor’s functional calculus and derived categories (abstract) (slides)
J. L. Taylor’s functional calculus theorem (1970) asserts that every commuting n-tuple T = (T1, . . . , Tn) of bounded linear operators on a Banach space E admits a holomorphic functional calculus on any neighborhood U of the joint spectrum ¾(T). This means that there exists a continuous homomorphism ° : O(U) ! B(E) (where O(U) is the algebra of holomorphic functions on U and B(E) is the algebra of bounded linear operators on E) that takes the coordinates z1, . . . , zn to T1, . . . , Tn, respectively. The original Taylor’s proof was quite involved. In 1972, Taylor developed a completely different and considerably shorter proof based on methods of Topological Homology. Later it was simplified and generalized by M. Putinar (1980) to the case of Fr´echet O(X)-modules, where X is a finite-dimensional Stein space. The idea of Taylor-Putinar’s construction is to establish an isomorphism between a Fr´echet O(X)-module M satisfying ¾(M) ½ U and the 0th cohomology of a certain double complex C of Fr´echet O(U)-modules. Unfortunately, C depends on the choice of a special cover of X by Stein open sets, and there seems to be no canonical way of associating C to M.

Our goal is to extend Taylor-Putinar’s theorem to the setting of derived categories. We believe that this is exactly the environment in which Taylor-Putinar’s
theorem is most naturally formulated and proved. Given an object M of the derived category D-(O(X)-mod) of Fr´echet O(X)-modules, we define the spectrum ¾(M) ½ X, and we show that for every open set U ½ X containing ¾(M) there is an isomorphism M »= R¡(U,OX) b­LO(X)M in D-(O(X)-mod). In the special case where M is a Fr´echet O(X)-module, this yields Taylor-Putinar’s result. Moreover, we have C = R¡(U,OX) b­LO(X)M, so C is natural in M when viewed as an object of the derived category.
Krzysztof Piszczek
Amenability properties of the noncommutative Schwartz space (slides)

The talk will be devoted to S, the so called noncommutative Schwartz space. This LMC Fr´echet ¤-algebra is a noncommutative analogue of the very important
Schwartz space, appearing naturally in the structure theory of Fr´echet spaces. The Schwartz space has also several natural representations as a space of functions. We will look at S from the viewpoint of the automatic continuity theory and we will examine how ‘amenable’ this algebra is. By the result of Pirkovskii we know S is not amenable. The main result of the talk will tell us S is not boundedly approximately amenable however it is approximately amenable.

Sutanu Roy
The Haagerup property of the generalised Drinfel'd double.
In this talk, we first discuss the generslised Drinfel'd double construction within the scope of modular or manageable multiplicative unitaries.
Our construction uses the following data: two C*-quantum groups and a bicharacter between them. Then we show that the generalised Drinfeld double has the Haagerup property whenever the underlying two quantum groups have the same. This shows that the Drinfel'd double of quantum ax+b, az+b and E(2)
groups has the Haagerup property.
Yuhei Suzuki
Amenable minimal Cantor systems of free groups arising from diagonal actions (slides)
We study amenable minimal Cantor systems of free groups. We show for every free group, (explicitly given) continuum many Kirchberg algebras are realized as the crossed product of an amenable minimal Cantor system of it. In particular this shows there are continuum many Kirchberg algebras such that each of which is decomposed tothe crossed products of amenable minimal Cantor systems of any virtually free group. We also give computations of K-groups for the diagonal actions of the boundary action and the odometer transformations. These computations with Matui's theorem classify their topological full groups.
Reiji Tomatsu
Product type actions of compact quantum groups (slides)
Abstract: A faithful product type action of the q-deformation of a connected semisimple compact Lie group is discussed.
Our main theorem states that such an action is induced from a minimal action
of the maximal torus. I will sketch out its proof.

Leonid Vainerman
Coamenability and quantum groupoids (slides)
We are discussing in which way the definitions and results on coamenable compact quantum groups can be extended to the framework of compact quantum groupoids. Some motivating examples are presented.
Ngai-Ching Wong
Isometries of real Hilbert C*-modules (slides)
Let T be a surjective real linear isometry between full real Hilbert C*-modules V and W, over real C*-algebras A and B, respectively. We show that the following conditions are equivalent.
(a) T is a 2-isometry;
(b) T is a complete isometry;
(c) T preserves ternary products;
(d) T preserves inner products;
(e) T is a module map.
When A and B are commutative, all these five conditions hold automatically.
This is a joint work with Ming-Hsiu Hsu.
Yong Zhang
Amenability properties of Banach algebra valued continuous functions (slides)
Let X be a compact Hausdorff space and A a Banach algebra. Then, with pointwise operations and the uniform norm, the space C(X,A) of all A-valued continuous functions is a Banach algebra. We investigate amenability, weakly amenability and generalized amenability of this algebra.